Theorem cplxpv1 ≪ | index | src | ≫

theorem cplxpv1 (A: set): $ Compl (Xp _V A) == Xp _V (Compl A) $;

Step	Hyp	Ref	Expression
1		bitr	(p e. Compl (Xp _V A) <-> ~p e. Xp _V A) -> (~p e. Xp _V A <-> p e. Xp _V (Compl A)) -> (p e. Compl (Xp _V A) <-> p e. Xp _V (Compl A))
2		elcpl	p e. Compl (Xp _V A) <-> ~p e. Xp _V A
3	1, 2	ax_mp	(~p e. Xp _V A <-> p e. Xp _V (Compl A)) -> (p e. Compl (Xp _V A) <-> p e. Xp _V (Compl A))
4		bitr4	(~p e. Xp _V A <-> ~snd p e. A) -> (p e. Xp _V (Compl A) <-> ~snd p e. A) -> (~p e. Xp _V A <-> p e. Xp _V (Compl A))
5		noteq	(p e. Xp _V A <-> snd p e. A) -> (~p e. Xp _V A <-> ~snd p e. A)
6		bitr	(p e. Xp _V A <-> fst p e. _V /\ snd p e. A) -> (fst p e. _V /\ snd p e. A <-> snd p e. A) -> (p e. Xp _V A <-> snd p e. A)
7		elxp	p e. Xp _V A <-> fst p e. _V /\ snd p e. A
8	6, 7	ax_mp	(fst p e. _V /\ snd p e. A <-> snd p e. A) -> (p e. Xp _V A <-> snd p e. A)
9		bian1	fst p e. _V -> (fst p e. _V /\ snd p e. A <-> snd p e. A)
10		elv	fst p e. _V
11	9, 10	ax_mp	fst p e. _V /\ snd p e. A <-> snd p e. A
12	8, 11	ax_mp	p e. Xp _V A <-> snd p e. A
13	5, 12	ax_mp	~p e. Xp _V A <-> ~snd p e. A
14	4, 13	ax_mp	(p e. Xp _V (Compl A) <-> ~snd p e. A) -> (~p e. Xp _V A <-> p e. Xp _V (Compl A))
15		bitr	(p e. Xp _V (Compl A) <-> fst p e. _V /\ snd p e. Compl A) -> (fst p e. _V /\ snd p e. Compl A <-> ~snd p e. A) -> (p e. Xp _V (Compl A) <-> ~snd p e. A)
16		elxp	p e. Xp _V (Compl A) <-> fst p e. _V /\ snd p e. Compl A
17	15, 16	ax_mp	(fst p e. _V /\ snd p e. Compl A <-> ~snd p e. A) -> (p e. Xp _V (Compl A) <-> ~snd p e. A)
18		bitr	(fst p e. _V /\ snd p e. Compl A <-> snd p e. Compl A) -> (snd p e. Compl A <-> ~snd p e. A) -> (fst p e. _V /\ snd p e. Compl A <-> ~snd p e. A)
19		bian1	fst p e. _V -> (fst p e. _V /\ snd p e. Compl A <-> snd p e. Compl A)
20	19, 10	ax_mp	fst p e. _V /\ snd p e. Compl A <-> snd p e. Compl A
21	18, 20	ax_mp	(snd p e. Compl A <-> ~snd p e. A) -> (fst p e. _V /\ snd p e. Compl A <-> ~snd p e. A)
22		elcpl	snd p e. Compl A <-> ~snd p e. A
23	21, 22	ax_mp	fst p e. _V /\ snd p e. Compl A <-> ~snd p e. A
24	17, 23	ax_mp	p e. Xp _V (Compl A) <-> ~snd p e. A
25	14, 24	ax_mp	~p e. Xp _V A <-> p e. Xp _V (Compl A)
26	3, 25	ax_mp	p e. Compl (Xp _V A) <-> p e. Xp _V (Compl A)
27	26	eqri	Compl (Xp _V A) == Xp _V (Compl A)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)