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Theorem unssdif 3145
Description: Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
Assertion
Ref Expression
unssdif (AB) ⊆ (V ∖ ((V ∖ A) ∖ B))

Proof of Theorem unssdif
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 vex 2534 . . . . . . . 8 x V
2 eldif 2900 . . . . . . . 8 (x (V ∖ A) ↔ (x V ¬ x A))
31, 2mpbiran 833 . . . . . . 7 (x (V ∖ A) ↔ ¬ x A)
43anbi1i 434 . . . . . 6 ((x (V ∖ A) ¬ x B) ↔ (¬ x A ¬ x B))
5 eldif 2900 . . . . . 6 (x ((V ∖ A) ∖ B) ↔ (x (V ∖ A) ¬ x B))
6 ioran 656 . . . . . 6 (¬ (x A x B) ↔ (¬ x A ¬ x B))
74, 5, 63bitr4i 201 . . . . 5 (x ((V ∖ A) ∖ B) ↔ ¬ (x A x B))
87biimpi 113 . . . 4 (x ((V ∖ A) ∖ B) → ¬ (x A x B))
98con2i 545 . . 3 ((x A x B) → ¬ x ((V ∖ A) ∖ B))
10 elun 3057 . . 3 (x (AB) ↔ (x A x B))
11 eldif 2900 . . . 4 (x (V ∖ ((V ∖ A) ∖ B)) ↔ (x V ¬ x ((V ∖ A) ∖ B)))
121, 11mpbiran 833 . . 3 (x (V ∖ ((V ∖ A) ∖ B)) ↔ ¬ x ((V ∖ A) ∖ B))
139, 10, 123imtr4i 190 . 2 (x (AB) → x (V ∖ ((V ∖ A) ∖ B)))
1413ssriv 2922 1 (AB) ⊆ (V ∖ ((V ∖ A) ∖ B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 616   wcel 1370  Vcvv 2531  cdif 2887  cun 2888  wss 2890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904
This theorem is referenced by: (None)
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