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Theorem unssdif 3199
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 2577 . . . . . . . 8 𝑥 ∈ V
2 eldif 2954 . . . . . . . 8 (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐴))
31, 2mpbiran 858 . . . . . . 7 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
43anbi1i 439 . . . . . 6 ((𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 eldif 2954 . . . . . 6 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥𝐵))
6 ioran 679 . . . . . 6 (¬ (𝑥𝐴𝑥𝐵) ↔ (¬ 𝑥𝐴 ∧ ¬ 𝑥𝐵))
74, 5, 63bitr4i 205 . . . . 5 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
87biimpi 117 . . . 4 (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) → ¬ (𝑥𝐴𝑥𝐵))
98con2i 567 . . 3 ((𝑥𝐴𝑥𝐵) → ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
10 elun 3111 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
11 eldif 2954 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)))
121, 11mpbiran 858 . . 3 (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
139, 10, 123imtr4i 194 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)))
1413ssriv 2976 1 (𝐴𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101  wo 639  wcel 1409  Vcvv 2574  cdif 2941  cun 2942  wss 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958
This theorem is referenced by: (None)
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