|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 8526 | . . 3 ⊢ 1o ≠ ∅ | |
| 2 | 1 | necomi 2995 | . 2 ⊢ ∅ ≠ 1o | 
| 3 | xpsndisj 6183 | . 2 ⊢ (∅ ≠ 1o → ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅) | |
| 4 | 2, 3 | ax-mp 5 | 1 ⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ≠ wne 2940 ∩ cin 3950 ∅c0 4333 {csn 4626 × cxp 5683 1oc1o 8499 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-suc 6390 df-1o 8506 | 
| This theorem is referenced by: endisj 9098 | 
| Copyright terms: Public domain | W3C validator |