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Mirrors > Home > MPE Home > Th. List > xpeq0 | Structured version Visualization version GIF version |
Description: At least one member of an empty Cartesian product is empty. (Contributed by NM, 27-Aug-2006.) |
Ref | Expression |
---|---|
xpeq0 | ⊢ ((𝐴 × 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpnz 6019 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | |
2 | 1 | necon2bbii 3070 | . 2 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
3 | ianor 978 | . 2 ⊢ (¬ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (¬ 𝐴 ≠ ∅ ∨ ¬ 𝐵 ≠ ∅)) | |
4 | nne 3023 | . . 3 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅) | |
5 | nne 3023 | . . 3 ⊢ (¬ 𝐵 ≠ ∅ ↔ 𝐵 = ∅) | |
6 | 4, 5 | orbi12i 911 | . 2 ⊢ ((¬ 𝐴 ≠ ∅ ∨ ¬ 𝐵 ≠ ∅) ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)) |
7 | 2, 3, 6 | 3bitri 299 | 1 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ≠ wne 3019 ∅c0 4294 × cxp 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-rel 5565 df-cnv 5566 |
This theorem is referenced by: xpcan 6036 xpcan2 6037 frxp 7823 rankxplim3 9313 xpcbas 17431 metn0 22973 hashxpe 30532 filnetlem4 33733 |
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