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Theorem inssun 3171
 Description: Intersection in terms of class difference and union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
Assertion
Ref Expression
inssun (AB) ⊆ (V ∖ ((V ∖ A) ∪ (V ∖ B)))

Proof of Theorem inssun
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 pm3.1 670 . . . . 5 ((x A x B) → ¬ (¬ x A ¬ x B))
2 eldifn 3061 . . . . . 6 (x (V ∖ A) → ¬ x A)
3 eldifn 3061 . . . . . 6 (x (V ∖ B) → ¬ x B)
42, 3orim12i 675 . . . . 5 ((x (V ∖ A) x (V ∖ B)) → (¬ x A ¬ x B))
51, 4nsyl 558 . . . 4 ((x A x B) → ¬ (x (V ∖ A) x (V ∖ B)))
6 elun 3078 . . . 4 (x ((V ∖ A) ∪ (V ∖ B)) ↔ (x (V ∖ A) x (V ∖ B)))
75, 6sylnibr 601 . . 3 ((x A x B) → ¬ x ((V ∖ A) ∪ (V ∖ B)))
8 elin 3120 . . 3 (x (AB) ↔ (x A x B))
9 vex 2554 . . . 4 x V
10 eldif 2921 . . . 4 (x (V ∖ ((V ∖ A) ∪ (V ∖ B))) ↔ (x V ¬ x ((V ∖ A) ∪ (V ∖ B))))
119, 10mpbiran 846 . . 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 2943 1 (AB) ⊆ (V ∖ ((V ∖ A) ∪ (V ∖ B)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ∨ wo 628   ∈ wcel 1390  Vcvv 2551   ∖ cdif 2908   ∪ cun 2909   ∩ cin 2910   ⊆ wss 2911 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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925 This theorem is referenced by: (None)
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