![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9612 | . 2 ⊢ ¬ 𝐴 ∈ 𝐴 | |
2 | disjsn 4711 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | mpbir 230 | 1 ⊢ (𝐴 ∩ {𝐴}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 ∩ cin 3943 ∅c0 4318 {csn 4624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-pr 5423 ax-reg 9607 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-nul 4319 df-sn 4625 df-pr 4627 |
This theorem is referenced by: bnj927 34336 bnj535 34457 sucdifsn2 37643 ressucdifsn2 37649 |
Copyright terms: Public domain | W3C validator |