Theorem insunsn ≪ | index | src | ≫

theorem insunsn (a b: nat): $ a ; b == sn a u. b $;

Step	Hyp	Ref	Expression
1		elins	x e. a ; b <-> x = a \/ x e. b
2		elun	x e. sn a u. b <-> x e. sn a \/ x e. b
3		oreq1	(x = a <-> x e. sn a) -> (x = a \/ x e. b <-> x e. sn a \/ x e. b)
4		bicom	(x e. sn a <-> x = a) -> (x = a <-> x e. sn a)
5		elsn	x e. sn a <-> x = a
6	4, 5	ax_mp	x = a <-> x e. sn a
7	3, 6	ax_mp	x = a \/ x e. b <-> x e. sn a \/ x e. b
8	1, 2, 7	bitr4gi	x e. a ; b <-> x e. sn a u. b
9	8	eqri	a ; b == sn a u. b

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)