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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 33692 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 7913 | . . 3 ⊢ 1o ≠ ∅ | |
2 | 1 | necomi 3015 | . 2 ⊢ ∅ ≠ 1o |
3 | disjsn2 4516 | . 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ({∅} ∩ {1o}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ≠ wne 2961 ∩ cin 3824 ∅c0 4173 {csn 4435 1oc1o 7890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 ax-nul 5061 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-v 3411 df-dif 3828 df-un 3830 df-in 3832 df-nul 4174 df-sn 4436 df-suc 6029 df-1o 7897 |
This theorem is referenced by: bj-2upln1upl 33789 |
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