Theorem xp0OLD 6156

Description: Obsolete version of xp0 5762 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 5761	. . 3 ⊢ (∅ × 𝐴) = ∅
2	1	cnveqi 5861	. 2 ⊢ ◡(∅ × 𝐴) = ◡∅
3		cnvxp 6155	. 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅)
4		cnv0 5870	. 2 ⊢ ◡∅ = ∅
5	2, 3, 4	3eqtr3i 2800	1 ⊢ (𝐴 × ∅) = ∅

Syntax hints:   = wceq 1567  ∅c0 4294   × cxp 5660  ^◡ccnv 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670

This theorem is referenced by: (None)