Theorem xp0OLD 6116

Description: Obsolete version of xp0 5724 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 5723	. . 3 ⊢ (∅ × 𝐴) = ∅
2	1	cnveqi 5823	. 2 ⊢ ◡(∅ × 𝐴) = ◡∅
3		cnvxp 6115	. 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅)
4		cnv0 6097	. 2 ⊢ ◡∅ = ∅
5	2, 3, 4	3eqtr3i 2768	1 ⊢ (𝐴 × ∅) = ∅

Syntax hints:   = wceq 1542  ∅c0 4274   × cxp 5622  ^◡ccnv 5623

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 2709  ax-sep 5231  ax-pr 5370

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 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632

This theorem is referenced by: (None)