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| Description: A class is disjoint from its singleton. A consequence of regularity. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 4-Apr-2019.) | 
| Ref | Expression | 
|---|---|
| disjcsn | ⊢ (𝐴 ∩ {𝐴}) = ∅ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elirr 9637 | . 2 ⊢ ¬ 𝐴 ∈ 𝐴 | |
| 2 | disjsn 4711 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
| 3 | 1, 2 | mpbir 231 | 1 ⊢ (𝐴 ∩ {𝐴}) = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ∅c0 4333 {csn 4626 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-reg 9632 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-nul 4334 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: bnj927 34783 bnj535 34904 sucdifsn2 38239 ressucdifsn2 38245 | 
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