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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 33131 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 7663 | . . 3 ⊢ 1𝑜 ≠ ∅ | |
2 | 1 | necomi 2918 | . 2 ⊢ ∅ ≠ 1𝑜 |
3 | disjsn2 4322 | . 2 ⊢ (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ({∅} ∩ {1𝑜}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1564 ≠ wne 2864 ∩ cin 3647 ∅c0 3991 {csn 4253 1𝑜c1o 7641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-nul 4865 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-ral 2987 df-v 3274 df-dif 3651 df-un 3653 df-in 3655 df-nul 3992 df-sn 4254 df-suc 5810 df-1o 7648 |
This theorem is referenced by: bj-2upln1upl 33207 |
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