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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 4260. (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 4216 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) | |
2 | eldif 3959 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
3 | 2 | notbii 320 | . . . . . 6 ⊢ (¬ 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) |
4 | 3 | anbi2i 624 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
5 | elin 3965 | . . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
6 | abai 826 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))) | |
7 | iman 403 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
8 | 7 | anbi2i 624 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
9 | 5, 6, 8 | 3bitri 297 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
10 | 4, 9 | bitr4i 278 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ 𝑥 ∈ (𝐵 ∩ 𝐴)) |
11 | 10 | difeqri 4125 | . . 3 ⊢ (𝐵 ∖ (𝐵 ∖ 𝐴)) = (𝐵 ∩ 𝐴) |
12 | 11 | eqeq1i 2738 | . 2 ⊢ ((𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
13 | 1, 12 | bitr4i 278 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-in 3956 df-ss 3966 |
This theorem is referenced by: ssdifim 4263 dfin4 4268 sscon34b 4295 sorpsscmpl 7724 sbthlem3 9085 fin23lem7 10311 fin23lem11 10312 compsscnvlem 10365 compssiso 10369 isf34lem4 10372 efgmnvl 19582 frlmlbs 21352 isopn2 22536 iincld 22543 iuncld 22549 clsval2 22554 ntrval2 22555 ntrdif 22556 clsdif 22557 cmclsopn 22566 opncldf1 22588 indiscld 22595 mretopd 22596 restcld 22676 pnrmopn 22847 conndisj 22920 hausllycmp 22998 kqcldsat 23237 filufint 23424 cfinufil 23432 ufilen 23434 alexsublem 23548 bcth3 24848 inmbl 25059 iccmbl 25083 mbfimaicc 25148 i1fd 25198 itgss3 25332 difuncomp 31785 iundifdifd 31793 iundifdif 31794 supppreima 31913 pmtrcnelor 32252 ist0cld 32813 cldssbrsiga 33185 unelcarsg 33311 kur14lem4 34200 cldbnd 35211 clsun 35213 mblfinlem3 36527 mblfinlem4 36528 ismblfin 36529 itg2addnclem 36539 fdc 36613 dssmapnvod 42771 ntrclsfveq1 42811 ntrclsfveq 42813 ntrclsneine0lem 42815 ntrclsiso 42818 ntrclsk2 42819 ntrclskb 42820 ntrclsk3 42821 ntrclsk13 42822 ntrclsk4 42823 clsneiel2 42860 neicvgel2 42871 salincl 45040 salexct 45050 ovnsubadd2lem 45361 lincext2 47136 opncldeqv 47534 |
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