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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-disjsn01 | Structured version Visualization version GIF version |
Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 35189 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-disjsn01 | ⊢ ({∅} ∩ {1o}) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1n0 8349 | . . 3 ⊢ 1o ≠ ∅ | |
2 | 1 | necomi 2996 | . 2 ⊢ ∅ ≠ 1o |
3 | disjsn2 4652 | . 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ({∅} ∩ {1o}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ≠ wne 2941 ∩ cin 3891 ∅c0 4262 {csn 4565 1oc1o 8321 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 ax-nul 5239 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-nul 4263 df-sn 4566 df-suc 6287 df-1o 8328 |
This theorem is referenced by: (None) |
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