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Mirrors > Home > ILE Home > Th. List > xp01disjl | GIF version |
Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.) |
Ref | Expression |
---|---|
xp01disjl | ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1n0 6329 | . . 3 ⊢ 1o ≠ ∅ | |
2 | 1 | necomi 2393 | . 2 ⊢ ∅ ≠ 1o |
3 | disjsn2 3586 | . 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
4 | xpdisj1 4963 | . 2 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅) | |
5 | 2, 3, 4 | mp2b 8 | 1 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ≠ wne 2308 ∩ cin 3070 ∅c0 3363 {csn 3527 × cxp 4537 1oc1o 6306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-suc 4293 df-xp 4545 df-rel 4546 df-1o 6313 |
This theorem is referenced by: djucomen 7072 djuassen 7073 xpdjuen 7074 |
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