Theorem xpeq0r 4689

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 4302	. . 3 ⊢ (A = ∅ → (A × B) = (∅ × B))
2		0xp 4363	. . 3 ⊢ (∅ × B) = ∅
3	1, 2	syl6eq 2085	. 2 ⊢ (A = ∅ → (A × B) = ∅)
4		xpeq2 4303	. . 3 ⊢ (B = ∅ → (A × B) = (A × ∅))
5		xp0 4686	. . 3 ⊢ (A × ∅) = ∅
6	4, 5	syl6eq 2085	. 2 ⊢ (B = ∅ → (A × B) = ∅)
7	3, 6	jaoi 635	1 ⊢ ((A = ∅ ∨ B = ∅) → (A × B) = ∅)

Syntax hints:   → wi 4   ∨ wo 628   = wceq 1242  ∅c0 3218   × cxp 4286

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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296

This theorem is referenced by: (None)