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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 6158 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | |
2 | 1 | necon2bbii 2992 | . 2 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) |
3 | ianor 980 | . 2 ⊢ (¬ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (¬ 𝐴 ≠ ∅ ∨ ¬ 𝐵 ≠ ∅)) | |
4 | nne 2944 | . . 3 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅) | |
5 | nne 2944 | . . 3 ⊢ (¬ 𝐵 ≠ ∅ ↔ 𝐵 = ∅) | |
6 | 4, 5 | orbi12i 913 | . 2 ⊢ ((¬ 𝐴 ≠ ∅ ∨ ¬ 𝐵 ≠ ∅) ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)) |
7 | 2, 3, 6 | 3bitri 296 | 1 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ≠ wne 2940 ∅c0 4322 × cxp 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 |
This theorem is referenced by: xpcan 6175 xpcan2 6176 frxp 8111 rankxplim3 9875 xpcbas 18129 metn0 23865 hashxpe 32014 filnetlem4 35261 |
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