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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 3821. (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 3779 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) | |
2 | eldif 3550 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
3 | 2 | notbii 309 | . . . . . 6 ⊢ (¬ 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) |
4 | 3 | anbi2i 726 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
5 | elin 3758 | . . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
6 | abai 832 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))) | |
7 | iman 439 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
8 | 7 | anbi2i 726 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
9 | 5, 6, 8 | 3bitri 285 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))) |
10 | 4, 9 | bitr4i 266 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ 𝑥 ∈ (𝐵 ∩ 𝐴)) |
11 | 10 | difeqri 3692 | . . 3 ⊢ (𝐵 ∖ (𝐵 ∖ 𝐴)) = (𝐵 ∩ 𝐴) |
12 | 11 | eqeq1i 2615 | . 2 ⊢ ((𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
13 | 1, 12 | bitr4i 266 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 |
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-v 3175 df-dif 3543 df-in 3547 df-ss 3554 |
This theorem is referenced by: dfin4 3826 sorpsscmpl 6846 sbthlem3 7957 fin23lem7 9021 fin23lem11 9022 compsscnvlem 9075 compssiso 9079 isf34lem4 9082 efgmnvl 17950 frlmlbs 19955 isopn2 20646 iincld 20653 iuncld 20659 clsval2 20664 ntrval2 20665 ntrdif 20666 clsdif 20667 cmclsopn 20676 opncldf1 20698 indiscld 20705 mretopd 20706 restcld 20786 pnrmopn 20957 conndisj 21029 hausllycmp 21107 kqcldsat 21346 filufint 21534 cfinufil 21542 ufilen 21544 alexsublem 21658 bcth3 22936 inmbl 23117 iccmbl 23141 mbfimaicc 23206 i1fd 23254 itgss3 23387 frgrawopreg2 26578 difuncomp 28752 iundifdifd 28762 iundifdif 28763 cldssbrsiga 29577 unelcarsg 29701 kur14lem4 30445 cldbnd 31491 clsun 31493 mblfinlem3 32618 mblfinlem4 32619 ismblfin 32620 itg2addnclem 32631 fdc 32711 dssmapnvod 37334 sscon34b 37337 ntrclsfveq1 37378 ntrclsfveq 37380 ntrclsneine0lem 37382 ntrclsiso 37385 ntrclsk2 37386 ntrclskb 37387 ntrclsk3 37388 ntrclsk13 37389 ntrclsk4 37390 clsneiel2 37427 neicvgel2 37438 salincl 39219 salexct 39228 ovnsubadd2lem 39535 frgrwopreg2 41488 lincext2 42038 |
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