Theorem xp0OLD 6158

Description: Obsolete version of xp0 5763 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 5762	. . 3 ⊢ (∅ × 𝐴) = ∅
2	1	cnveqi 5862	. 2 ⊢ ◡(∅ × 𝐴) = ◡∅
3		cnvxp 6156	. 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅)
4		cnv0 5871	. 2 ⊢ ◡∅ = ∅
5	2, 3, 4	3eqtr3i 2796	1 ⊢ (𝐴 × ∅) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4286   × cxp 5661  ^◡ccnv 5662

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671

This theorem is used by: (None)