Theorem xpeq0r 4673

Description: A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)

Assertion

Ref	Expression
xpeq0r	⊢ ((A = ∅ ∨ B = ∅) → (A × B) = ∅)

Proof of Theorem xpeq0r

Step	Hyp	Ref	Expression
1		xpeq1 4286	. . 3 ⊢ (A = ∅ → (A × B) = (∅ × B))
2		0xp 4347	. . 3 ⊢ (∅ × B) = ∅
3	1, 2	syl6eq 2070	. 2 ⊢ (A = ∅ → (A × B) = ∅)
4		xpeq2 4287	. . 3 ⊢ (B = ∅ → (A × B) = (A × ∅))
5		xp0 4670	. . 3 ⊢ (A × ∅) = ∅
6	4, 5	syl6eq 2070	. 2 ⊢ (B = ∅ → (A × B) = ∅)
7	3, 6	jaoi 623	1 ⊢ ((A = ∅ ∨ B = ∅) → (A × B) = ∅)

Syntax hints:   → wi 4   ∨ wo 616   = wceq 1228  ∅c0 3201   × cxp 4270

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280

This theorem is referenced by: (None)