Theorem dfss4 4218

Description: Subclass defined in terms of class difference. See comments under dfun2 4219. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
dfss4	⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴)

Proof of Theorem dfss4

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseqin2 4172	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
2		eldif 3908	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
3	2	notbii 320	. . . . . 6 ⊢ (¬ 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
4	3	anbi2i 623	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 ∧ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
5		elin 3914	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
6		abai 826	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)))
7		iman 401	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
8	7	anbi2i 623	. . . . . 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 4077	. . 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)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-in 3905  df-ss 3915

This theorem is referenced by:  ssdifim  4222  dfin4  4227  sscon34b  4253  sorpsscmpl  7676  sbthlem3  9013  fin23lem7  10218  fin23lem11  10219  compsscnvlem  10272  compssiso  10276  isf34lem4  10279  efgmnvl  19634  frlmlbs  21743  isopn2  22967  iincld  22974  iuncld  22980  clsval2  22985  ntrval2  22986  ntrdif  22987  clsdif  22988  cmclsopn  22997  opncldf1  23019  indiscld  23026  mretopd  23027  restcld  23107  pnrmopn  23278  conndisj  23351  hausllycmp  23429  kqcldsat  23668  filufint  23855  cfinufil  23863  ufilen  23865  alexsublem  23979  bcth3  25278  inmbl  25490  iccmbl  25514  mbfimaicc  25579  i1fd  25629  itgss3  25763  difuncomp  32554  iundifdifd  32562  iundifdif  32563  supppreima  32696  pmtrcnelor  33101  evlextv  33635  ist0cld  33918  cldssbrsiga  34272  unelcarsg  34397  kur14lem4  35325  cldbnd  36442  clsun  36444  mblfinlem3  37772  mblfinlem4  37773  ismblfin  37774  itg2addnclem  37784  fdc  37858  dssmapnvod  44177  ntrclsfveq1  44217  ntrclsfveq  44219  ntrclsneine0lem  44221  ntrclsiso  44224  ntrclsk2  44225  ntrclskb  44226  ntrclsk3  44227  ntrclsk13  44228  ntrclsk4  44229  clsneiel2  44266  neicvgel2  44277  salincl  46484  salexct  46494  ovnsubadd2lem  46805  lincext2  48617  opncldeqv  49063