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Theorem unssin 3153
 Description: Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
Assertion
Ref Expression
unssin (AB) ⊆ (V ∖ ((V ∖ A) ∩ (V ∖ B)))

Proof of Theorem unssin
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 oranim 800 . . . . 5 ((x A x B) → ¬ (¬ x A ¬ x B))
2 eldifn 3044 . . . . . 6 (x (V ∖ A) → ¬ x A)
3 eldifn 3044 . . . . . 6 (x (V ∖ B) → ¬ x B)
42, 3anim12i 321 . . . . 5 ((x (V ∖ A) x (V ∖ B)) → (¬ x A ¬ x B))
51, 4nsyl 546 . . . 4 ((x A x B) → ¬ (x (V ∖ A) x (V ∖ B)))
6 elin 3103 . . . 4 (x ((V ∖ A) ∩ (V ∖ B)) ↔ (x (V ∖ A) x (V ∖ B)))
75, 6sylnibr 589 . . 3 ((x A x B) → ¬ x ((V ∖ A) ∩ (V ∖ B)))
8 elun 3061 . . 3 (x (AB) ↔ (x A x B))
9 vex 2538 . . . 4 x V
10 eldif 2904 . . . 4 (x (V ∖ ((V ∖ A) ∩ (V ∖ B))) ↔ (x V ¬ x ((V ∖ A) ∩ (V ∖ B))))
119, 10mpbiran 835 . . 3 (x (V ∖ ((V ∖ A) ∩ (V ∖ B))) ↔ ¬ x ((V ∖ A) ∩ (V ∖ B)))
127, 8, 113imtr4i 190 . 2 (x (AB) → x (V ∖ ((V ∖ A) ∩ (V ∖ B))))
1312ssriv 2926 1 (AB) ⊆ (V ∖ ((V ∖ A) ∩ (V ∖ B)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ∨ wo 616   ∈ wcel 1374  Vcvv 2535   ∖ cdif 2891   ∪ cun 2892   ∩ cin 2893   ⊆ wss 2894 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908 This theorem is referenced by: (None)
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