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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 8102 | . . 3 ⊢ 1o ≠ ∅ | |
2 | 1 | necomi 3041 | . 2 ⊢ ∅ ≠ 1o |
3 | xpsndisj 5987 | . 2 ⊢ (∅ ≠ 1o → ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ≠ wne 2987 ∩ cin 3880 ∅c0 4243 {csn 4525 × cxp 5517 1oc1o 8078 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-suc 6165 df-1o 8085 |
This theorem is referenced by: endisj 8587 |
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