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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 6016 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | |
2 | 1 | necon2bbii 3067 | . 2 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
3 | ianor 978 | . 2 ⊢ (¬ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (¬ 𝐴 ≠ ∅ ∨ ¬ 𝐵 ≠ ∅)) | |
4 | nne 3020 | . . 3 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅) | |
5 | nne 3020 | . . 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 1537 ≠ wne 3016 ∅c0 4291 × cxp 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 |
This theorem is referenced by: xpcan 6033 xpcan2 6034 frxp 7820 rankxplim3 9310 xpcbas 17428 metn0 22970 hashxpe 30529 filnetlem4 33729 |
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