Theorem prcnv ≪ | index | src | ≫

theorem prcnv (A: set) (x y: nat): $ x, y e. cnv A <-> y, x e. A $;

Step	Hyp	Ref	Expression
1		id	_2 = y -> _2 = y
2	1	anwr	_1 = x /\ _2 = y -> _2 = y
3		id	_1 = x -> _1 = x
4	3	anwl	_1 = x /\ _2 = y -> _1 = x
5	2, 4	preqd	_1 = x /\ _2 = y -> _2, _1 = y, x
6		eqsidd	_1 = x /\ _2 = y -> A == A
7	5, 6	eleqd	_1 = x /\ _2 = y -> (_2, _1 e. A <-> y, x e. A)
8	7	elabed	_1 = x -> (y e. {_2 \| _2, _1 e. A} <-> y, x e. A)
9	8	elsabe	x, y e. S\ _1, {_2 \| _2, _1 e. A} <-> y, x e. A
10	9	conv cnv	x, y e. cnv A <-> y, x e. 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)