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Mirrors > Home > MPE Home > Th. List > disj3 | Structured version Visualization version GIF version |
Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.) |
Ref | Expression |
---|---|
disj3 | ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.71 660 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) | |
2 | eldif 3550 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
3 | 2 | bibi2i 326 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) |
4 | 1, 3 | bitr4i 266 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) |
5 | 4 | albii 1737 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) |
6 | disj1 3971 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | |
7 | dfcleq 2604 | . 2 ⊢ (𝐴 = (𝐴 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) | |
8 | 5, 6, 7 | 3bitr4i 291 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∩ cin 3539 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-dif 3543 df-in 3547 df-nul 3875 |
This theorem is referenced by: disjel 3975 disj4 3977 uneqdifeq 4009 uneqdifeqOLD 4010 difprsn1 4271 diftpsn3 4273 diftpsn3OLD 4274 ssunsn2 4299 orddif 5737 php 8029 hartogslem1 8330 infeq5i 8416 cantnfp1lem3 8460 cda1dif 8881 infcda1 8898 ssxr 9986 dprd2da 18264 dmdprdsplit2lem 18267 ablfac1eulem 18294 lbsextlem4 18982 opsrtoslem2 19306 alexsublem 21658 volun 23120 lhop1lem 23580 ex-dif 26672 difeq 28739 imadifxp 28796 disjdsct 28863 carsgclctunlem1 29706 probun 29808 ballotlemfp1 29880 topdifinfeq 32374 finixpnum 32564 poimirlem11 32590 poimirlem12 32591 poimirlem13 32592 poimirlem14 32593 poimirlem16 32595 poimirlem18 32597 poimirlem21 32600 poimirlem22 32601 poimirlem27 32606 asindmre 32665 kelac2 36653 pwfi2f1o 36684 iccdifioo 38588 iccdifprioo 38589 |
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