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Theorem inssun 3177
 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 pm3.1 671 . . . . 5 ((𝑥𝐴𝑥𝐵) → ¬ (¬ 𝑥𝐴 ∨ ¬ 𝑥𝐵))
2 eldifn 3067 . . . . . 6 (𝑥 ∈ (V ∖ 𝐴) → ¬ 𝑥𝐴)
3 eldifn 3067 . . . . . 6 (𝑥 ∈ (V ∖ 𝐵) → ¬ 𝑥𝐵)
42, 3orim12i 676 . . . . 5 ((𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)) → (¬ 𝑥𝐴 ∨ ¬ 𝑥𝐵))
51, 4nsyl 558 . . . 4 ((𝑥𝐴𝑥𝐵) → ¬ (𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)))
6 elun 3084 . . . 4 (𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) ↔ (𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)))
75, 6sylnibr 602 . . 3 ((𝑥𝐴𝑥𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
8 elin 3126 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
9 vex 2560 . . . 4 𝑥 ∈ V
10 eldif 2927 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))))
119, 10mpbiran 847 . . 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 2949 1 (𝐴𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ∨ wo 629   ∈ wcel 1393  Vcvv 2557   ∖ cdif 2914   ∪ cun 2915   ∩ cin 2916   ⊆ wss 2917 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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