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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-disjcsn | Structured version Visualization version GIF version |
Description: A class is disjoint from its singleton. A consequence of regularity. Shorter proof than bnj521 32007 and does not depend on df-ne 3017. (Contributed by BJ, 4-Apr-2019.) |
Ref | Expression |
---|---|
bj-disjcsn | ⊢ (𝐴 ∩ {𝐴}) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 9061 | . 2 ⊢ ¬ 𝐴 ∈ 𝐴 | |
2 | disjsn 4647 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | mpbir 233 | 1 ⊢ (𝐴 ∩ {𝐴}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ∅c0 4291 {csn 4567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-reg 9056 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-nul 4292 df-sn 4568 df-pr 4570 |
This theorem is referenced by: (None) |
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