Theorem xp0OLD 6148

Description: Obsolete version of xp0 5750 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 5749	. . 3 ⊢ (∅ × 𝐴) = ∅
2	1	cnveqi 5851	. 2 ⊢ ◡(∅ × 𝐴) = ◡∅
3		cnvxp 6146	. 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅)
4		cnv0 5860	. 2 ⊢ ◡∅ = ∅
5	2, 3, 4	3eqtr3i 2791	1 ⊢ (𝐴 × ∅) = ∅

This proof depends on syntax axioms:   = wceq 1569  ∅c0 4278   × cxp 5648  ^◡ccnv 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658

This theorem is used by: (None)