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Mirrors > Home > ILE Home > Th. List > xp01disj | 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 6485 | . . 3 ⊢ 1o ≠ ∅ | |
2 | 1 | necomi 2449 | . 2 ⊢ ∅ ≠ 1o |
3 | xpsndisj 5092 | . 2 ⊢ (∅ ≠ 1o → ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ≠ wne 2364 ∩ cin 3152 ∅c0 3446 {csn 3618 × cxp 4657 1oc1o 6462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-v 2762 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-br 4030 df-opab 4091 df-suc 4402 df-xp 4665 df-rel 4666 df-cnv 4667 df-1o 6469 |
This theorem is referenced by: endisj 6878 |
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