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Mirrors > Home > MPE Home > Th. List > dfss4 | Structured version Visualization version GIF version |
Description: Subclass defined in terms of class difference. See comments under dfun2 4235. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
dfss4 | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqin2 4191 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) | |
2 | eldif 3945 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
3 | 2 | notbii 321 | . . . . . 6 ⊢ (¬ 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) |
4 | 3 | anbi2i 622 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
5 | elin 4168 | . . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
6 | abai 822 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))) | |
7 | iman 402 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
8 | 7 | anbi2i 622 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
9 | 5, 6, 8 | 3bitri 298 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
10 | 4, 9 | bitr4i 279 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ 𝑥 ∈ (𝐵 ∩ 𝐴)) |
11 | 10 | difeqri 4100 | . . 3 ⊢ (𝐵 ∖ (𝐵 ∖ 𝐴)) = (𝐵 ∩ 𝐴) |
12 | 11 | eqeq1i 2826 | . 2 ⊢ ((𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
13 | 1, 12 | bitr4i 279 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 |
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-rab 3147 df-v 3497 df-dif 3938 df-in 3942 df-ss 3951 |
This theorem is referenced by: ssdifim 4238 dfin4 4243 sorpsscmpl 7449 sbthlem3 8618 fin23lem7 9727 fin23lem11 9728 compsscnvlem 9781 compssiso 9785 isf34lem4 9788 efgmnvl 18771 frlmlbs 20871 isopn2 21570 iincld 21577 iuncld 21583 clsval2 21588 ntrval2 21589 ntrdif 21590 clsdif 21591 cmclsopn 21600 opncldf1 21622 indiscld 21629 mretopd 21630 restcld 21710 pnrmopn 21881 conndisj 21954 hausllycmp 22032 kqcldsat 22271 filufint 22458 cfinufil 22466 ufilen 22468 alexsublem 22582 bcth3 23863 inmbl 24072 iccmbl 24096 mbfimaicc 24161 i1fd 24211 itgss3 24344 difuncomp 30233 iundifdifd 30242 iundifdif 30243 pmtrcnelor 30663 cldssbrsiga 31346 unelcarsg 31470 kur14lem4 32354 cldbnd 33572 clsun 33574 mblfinlem3 34813 mblfinlem4 34814 ismblfin 34815 itg2addnclem 34825 fdc 34903 dssmapnvod 40246 sscon34b 40249 ntrclsfveq1 40290 ntrclsfveq 40292 ntrclsneine0lem 40294 ntrclsiso 40297 ntrclsk2 40298 ntrclskb 40299 ntrclsk3 40300 ntrclsk13 40301 ntrclsk4 40302 clsneiel2 40339 neicvgel2 40350 salincl 42489 salexct 42498 ovnsubadd2lem 42808 lincext2 44408 |
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