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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 6159 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | |
2 | 1 | necon2bbii 2993 | . 2 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
3 | ianor 981 | . 2 ⊢ (¬ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (¬ 𝐴 ≠ ∅ ∨ ¬ 𝐵 ≠ ∅)) | |
4 | nne 2945 | . . 3 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅) | |
5 | nne 2945 | . . 3 ⊢ (¬ 𝐵 ≠ ∅ ↔ 𝐵 = ∅) | |
6 | 4, 5 | orbi12i 914 | . 2 ⊢ ((¬ 𝐴 ≠ ∅ ∨ ¬ 𝐵 ≠ ∅) ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)) |
7 | 2, 3, 6 | 3bitri 297 | 1 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ≠ wne 2941 ∅c0 4323 × cxp 5675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 |
This theorem is referenced by: xpcan 6176 xpcan2 6177 frxp 8112 rankxplim3 9876 xpcbas 18130 metn0 23866 hashxpe 32019 filnetlem4 35266 |
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