Theorem xp0OLD 6160

Description: Obsolete version of xp0 5756 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 5755	. . 3 ⊢ (∅ × 𝐴) = ∅
2	1	cnveqi 5858	. 2 ⊢ ◡(∅ × 𝐴) = ◡∅
3		cnvxp 6158	. 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅)
4		cnv0 5867	. 2 ⊢ ◡∅ = ∅
5	2, 3, 4	3eqtr3i 2794	1 ⊢ (𝐴 × ∅) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4281   × cxp 5653  ^◡ccnv 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663

This theorem is used by: (None)