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Mirrors > Home > MPE Home > Th. List > xp01disj | Structured version Visualization version GIF version |
Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.) |
Ref | Expression |
---|---|
xp01disj | ⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1n0 8544 | . . 3 ⊢ 1o ≠ ∅ | |
2 | 1 | necomi 3001 | . 2 ⊢ ∅ ≠ 1o |
3 | xpsndisj 6194 | . 2 ⊢ (∅ ≠ 1o → ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ≠ wne 2946 ∩ cin 3975 ∅c0 4352 {csn 4648 × cxp 5698 1oc1o 8515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-xp 5706 df-rel 5707 df-cnv 5708 df-suc 6401 df-1o 8522 |
This theorem is referenced by: endisj 9124 |
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