Theorem xp0OLD 6155

Description: Obsolete version of xp0 5761 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 5760	. . 3 ⊢ (∅ × 𝐴) = ∅
2	1	cnveqi 5860	. 2 ⊢ ◡(∅ × 𝐴) = ◡∅
3		cnvxp 6154	. 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅)
4		cnv0 5869	. 2 ⊢ ◡∅ = ∅
5	2, 3, 4	3eqtr3i 2794	1 ⊢ (𝐴 × ∅) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4286   × cxp 5659  ^◡ccnv 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669

This theorem is used by: (None)