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Theorem undif3ss 3258
Description: A subset relationship involving class union and class difference. In classical logic, this would be equality rather than subset, as in the first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Jim Kingdon, 28-Jul-2018.)
Assertion
Ref Expression
undif3ss (𝐴 ∪ (𝐵𝐶)) ⊆ ((𝐴𝐵) ∖ (𝐶𝐴))

Proof of Theorem undif3ss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elun 3139 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
2 eldif 3006 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
32orbi2i 714 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4 orc 668 . . . . . . 7 (𝑥𝐴 → (𝑥𝐴𝑥𝐵))
5 olc 667 . . . . . . 7 (𝑥𝐴 → (¬ 𝑥𝐶𝑥𝐴))
64, 5jca 300 . . . . . 6 (𝑥𝐴 → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
7 olc 667 . . . . . . 7 (𝑥𝐵 → (𝑥𝐴𝑥𝐵))
8 orc 668 . . . . . . 7 𝑥𝐶 → (¬ 𝑥𝐶𝑥𝐴))
97, 8anim12i 331 . . . . . 6 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
106, 9jaoi 671 . . . . 5 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
11 simpl 107 . . . . . . 7 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → 𝑥𝐴)
1211orcd 687 . . . . . 6 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
13 olc 667 . . . . . 6 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
14 orc 668 . . . . . . 7 (𝑥𝐴 → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1514adantr 270 . . . . . 6 ((𝑥𝐴𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1614adantl 271 . . . . . 6 ((𝑥𝐵𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1712, 13, 15, 16ccase 910 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1810, 17impbii 124 . . . 4 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
191, 3, 183bitri 204 . . 3 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
20 elun 3139 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2120biimpri 131 . . . . 5 ((𝑥𝐴𝑥𝐵) → 𝑥 ∈ (𝐴𝐵))
22 pm4.53r 842 . . . . . 6 ((¬ 𝑥𝐶𝑥𝐴) → ¬ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
23 eldif 3006 . . . . . 6 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
2422, 23sylnibr 637 . . . . 5 ((¬ 𝑥𝐶𝑥𝐴) → ¬ 𝑥 ∈ (𝐶𝐴))
2521, 24anim12i 331 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
26 eldif 3006 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
2725, 26sylibr 132 . . 3 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
2819, 27sylbi 119 . 2 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) → 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
2928ssriv 3027 1 (𝐴 ∪ (𝐵𝐶)) ⊆ ((𝐴𝐵) ∖ (𝐶𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wo 664  wcel 1438  cdif 2994  cun 2995  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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