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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 7462 | . . 3 ⊢ 1𝑜 ≠ ∅ | |
2 | 1 | necomi 2836 | . 2 ⊢ ∅ ≠ 1𝑜 |
3 | xpsndisj 5476 | . 2 ⊢ (∅ ≠ 1𝑜 → ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ≠ wne 2780 ∩ cin 3539 ∅c0 3874 {csn 4125 × cxp 5036 1𝑜c1o 7440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-suc 5646 df-1o 7447 |
This theorem is referenced by: endisj 7932 uncdadom 8876 cdaun 8877 cdaen 8878 cda1dif 8881 pm110.643 8882 cdacomen 8886 cdaassen 8887 xpcdaen 8888 mapcdaen 8889 cdadom1 8891 infcda1 8898 |
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