Theorem xp0OLD 6126

Description: Obsolete version of xp0 5734 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 5733	. . 3 ⊢ (∅ × 𝐴) = ∅
2	1	cnveqi 5833	. 2 ⊢ ◡(∅ × 𝐴) = ◡∅
3		cnvxp 6125	. 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅)
4		cnv0 6107	. 2 ⊢ ◡∅ = ∅
5	2, 3, 4	3eqtr3i 2768	1 ⊢ (𝐴 × ∅) = ∅

Syntax hints:   = wceq 1542  ∅c0 4287   × cxp 5632  ^◡ccnv 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5640  df-rel 5641  df-cnv 5642

This theorem is referenced by: (None)