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Mirrors > Home > MPE Home > Th. List > disjcsn | Structured version Visualization version GIF version |
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 9635 | . 2 ⊢ ¬ 𝐴 ∈ 𝐴 | |
2 | disjsn 4716 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | mpbir 231 | 1 ⊢ (𝐴 ∩ {𝐴}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ∅c0 4339 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-pr 5438 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-nul 4340 df-sn 4632 df-pr 4634 |
This theorem is referenced by: bnj927 34762 bnj535 34883 sucdifsn2 38219 ressucdifsn2 38225 |
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