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Mirrors > Home > ILE Home > Th. List > xpeq0r | GIF version |
Description: A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.) |
Ref | Expression |
---|---|
xpeq0r | ⊢ ((A = ∅ ∨ B = ∅) → (A × B) = ∅) |
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) = ∅) |
Colors of variables: wff set class |
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-bndl 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) |
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