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| 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 6178 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) | |
| 2 | 1 | necon2bbii 2991 | . 2 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)) | 
| 3 | ianor 983 | . 2 ⊢ (¬ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (¬ 𝐴 ≠ ∅ ∨ ¬ 𝐵 ≠ ∅)) | |
| 4 | nne 2943 | . . 3 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅) | |
| 5 | nne 2943 | . . 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 206 ∧ wa 395 ∨ wo 847 = wceq 1539 ≠ wne 2939 ∅c0 4332 × cxp 5682 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 | 
| This theorem is referenced by: xpcan 6195 xpcan2 6196 frxp 8152 rankxplim3 9922 xpcbas 18224 metn0 24371 hashxpe 32812 filnetlem4 36383 fucofvalne 49043 | 
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