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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 7842 | . . 3 ⊢ 1o ≠ ∅ | |
2 | 1 | necomi 3053 | . 2 ⊢ ∅ ≠ 1o |
3 | xpsndisj 5798 | . 2 ⊢ (∅ ≠ 1o → ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ≠ wne 2999 ∩ cin 3797 ∅c0 4144 {csn 4397 × cxp 5340 1oc1o 7819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-xp 5348 df-rel 5349 df-cnv 5350 df-suc 5969 df-1o 7826 |
This theorem is referenced by: endisj 8316 uncdadom 9308 cdaun 9309 cdaen 9310 cda1dif 9313 pm110.643 9314 cdacomen 9318 cdaassen 9319 xpcdaen 9320 mapcdaen 9321 cdadom1 9323 infcda1 9330 |
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