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Theorem inssun 3237
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 pm3.1 706 . . . . 5 ((𝑥𝐴𝑥𝐵) → ¬ (¬ 𝑥𝐴 ∨ ¬ 𝑥𝐵))
2 eldifn 3121 . . . . . 6 (𝑥 ∈ (V ∖ 𝐴) → ¬ 𝑥𝐴)
3 eldifn 3121 . . . . . 6 (𝑥 ∈ (V ∖ 𝐵) → ¬ 𝑥𝐵)
42, 3orim12i 711 . . . . 5 ((𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)) → (¬ 𝑥𝐴 ∨ ¬ 𝑥𝐵))
51, 4nsyl 593 . . . 4 ((𝑥𝐴𝑥𝐵) → ¬ (𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)))
6 elun 3139 . . . 4 (𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) ↔ (𝑥 ∈ (V ∖ 𝐴) ∨ 𝑥 ∈ (V ∖ 𝐵)))
75, 6sylnibr 637 . . 3 ((𝑥𝐴𝑥𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
8 elin 3181 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
9 vex 2622 . . . 4 𝑥 ∈ V
10 eldif 3006 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))))
119, 10mpbiran 886 . . 3 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
127, 8, 113imtr4i 199 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))))
1312ssriv 3027 1 (𝐴𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wo 664  wcel 1438  Vcvv 2619  cdif 2994  cun 2995  cin 2996  wss 2997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010
This theorem is referenced by: (None)
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