Theorem xp0OLD 6122

Description: Obsolete version of xp0 5731 as of 1-Feb-2026. (Contributed by NM, 12-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
xp0OLD	⊢ (𝐴 × ∅) = ∅

Proof of Theorem xp0OLD

Step	Hyp	Ref	Expression
1		0xp 5730	. . 3 ⊢ (∅ × 𝐴) = ∅
2	1	cnveqi 5829	. 2 ⊢ ◡(∅ × 𝐴) = ◡∅
3		cnvxp 6121	. 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅)
4		cnv0 6103	. 2 ⊢ ◡∅ = ∅
5	2, 3, 4	3eqtr3i 2767	1 ⊢ (𝐴 × ∅) = ∅

Syntax hints:   = wceq 1542  ∅c0 4273   × cxp 5629  ^◡ccnv 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639

This theorem is referenced by: (None)