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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 | pm4.71 558 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) | |
2 | eldif 3945 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
3 | 2 | bibi2i 339 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) |
4 | 1, 3 | bitr4i 279 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) |
5 | 4 | albii 1811 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) |
6 | disj1 4399 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | |
7 | dfcleq 2815 | . 2 ⊢ (𝐴 = (𝐴 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) | |
8 | 5, 6, 7 | 3bitr4i 304 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 = wceq 1528 ∈ wcel 2105 ∖ cdif 3932 ∩ cin 3934 ∅c0 4290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3497 df-dif 3938 df-in 3942 df-nul 4291 |
This theorem is referenced by: disjel 4404 disj4 4406 uneqdifeq 4436 difprsn1 4727 diftpsn3 4729 ssunsn2 4754 orddif 6278 php 8690 hartogslem1 8995 infeq5i 9088 cantnfp1lem3 9132 dju1dif 9587 infdju1 9604 ssxr 10699 dprd2da 19095 dmdprdsplit2lem 19098 ablfac1eulem 19125 lbsextlem4 19864 opsrtoslem2 20195 alexsublem 22582 volun 24075 lhop1lem 24539 ex-dif 28130 difeq 30208 imadifxp 30280 disjdsct 30365 fzodif1 30443 carsgclctunlem1 31475 probun 31577 ballotlemfp1 31649 bj-disj2r 34238 topdifinfeq 34514 finixpnum 34759 lindsadd 34767 poimirlem11 34785 poimirlem12 34786 poimirlem13 34787 poimirlem14 34788 poimirlem16 34790 poimirlem18 34792 poimirlem21 34795 poimirlem22 34796 poimirlem27 34801 asindmre 34859 dffltz 39151 kelac2 39545 pwfi2f1o 39576 iccdifioo 41671 iccdifprioo 41672 |
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