Theorem xp0OLD 6105

Description: Obsolete version of xp0 5714 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 5713	. . 3 ⊢ (∅ × 𝐴) = ∅
2	1	cnveqi 5813	. 2 ⊢ ◡(∅ × 𝐴) = ◡∅
3		cnvxp 6104	. 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅)
4		cnv0 6086	. 2 ⊢ ◡∅ = ∅
5	2, 3, 4	3eqtr3i 2762	1 ⊢ (𝐴 × ∅) = ∅

Syntax hints:   = wceq 1541  ∅c0 4280   × cxp 5612  ^◡ccnv 5613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622

This theorem is referenced by: (None)