Theorem xp0 2652

Description: The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37.

Assertion

Ref	Expression
xp0	⊢ (A × ∅) = ∅

Proof of Theorem xp0

Step	Hyp	Ref	Expression
1		xp0r 2474	. . 3 ⊢ (∅ × A) = ∅
2		cnveq 2513	. . 3 ⊢ ((∅ × A) = ∅ → ◡(∅ × A) = ◡∅)
3	1, 2	ax-mp 6	. 2 ⊢ ◡(∅ × A) = ◡∅
4		cnvxp 2651	. 2 ⊢ ◡(∅ × A) = (A × ∅)
5		cnv0 2633	. 2 ⊢ ◡∅ = ∅
6	3, 4, 5	3eqtr3 1124	1 ⊢ (A × ∅) = ∅

Syntax hints:   = wceq 1091  ∅c0 1707   × cxp 2408  ^◡ccnv 2409

This theorem is referenced by:  xpdisj2 2654  aceq5lem3 3560  xpcdaen 3726  infxpidmlem4 4936

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426

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