Theorem xp0OLD 6113

Description: Obsolete version of xp0 5721 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 5720	. . 3 ⊢ (∅ × 𝐴) = ∅
2	1	cnveqi 5819	. 2 ⊢ ◡(∅ × 𝐴) = ◡∅
3		cnvxp 6112	. 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅)
4		cnv0 5828	. 2 ⊢ ◡∅ = ∅
5	2, 3, 4	3eqtr3i 2772	1 ⊢ (𝐴 × ∅) = ∅

Syntax hints:   = wceq 1548  ∅c0 4264   × cxp 5619  ^◡ccnv 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-11 2170  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-cnv 5629

This theorem is referenced by: (None)