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Theorem undif3ss 3243
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 3125 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
2 eldif 2993 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
32orbi2i 712 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4 orc 666 . . . . . . 7 (𝑥𝐴 → (𝑥𝐴𝑥𝐵))
5 olc 665 . . . . . . 7 (𝑥𝐴 → (¬ 𝑥𝐶𝑥𝐴))
64, 5jca 300 . . . . . 6 (𝑥𝐴 → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
7 olc 665 . . . . . . 7 (𝑥𝐵 → (𝑥𝐴𝑥𝐵))
8 orc 666 . . . . . . 7 𝑥𝐶 → (¬ 𝑥𝐶𝑥𝐴))
97, 8anim12i 331 . . . . . 6 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
106, 9jaoi 669 . . . . 5 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
11 simpl 107 . . . . . . 7 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → 𝑥𝐴)
1211orcd 685 . . . . . 6 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
13 olc 665 . . . . . 6 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
14 orc 666 . . . . . . 7 (𝑥𝐴 → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1514adantr 270 . . . . . 6 ((𝑥𝐴𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1614adantl 271 . . . . . 6 ((𝑥𝐵𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1712, 13, 15, 16ccase 906 . . . . 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 3125 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2120biimpri 131 . . . . 5 ((𝑥𝐴𝑥𝐵) → 𝑥 ∈ (𝐴𝐵))
22 pm4.53r 838 . . . . . 6 ((¬ 𝑥𝐶𝑥𝐴) → ¬ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
23 eldif 2993 . . . . . 6 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
2422, 23sylnibr 635 . . . . 5 ((¬ 𝑥𝐶𝑥𝐴) → ¬ 𝑥 ∈ (𝐶𝐴))
2521, 24anim12i 331 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
26 eldif 2993 . . . 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 3014 1 (𝐴 ∪ (𝐵𝐶)) ⊆ ((𝐴𝐵) ∖ (𝐶𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wo 662  wcel 1434  cdif 2981  cun 2982  wss 2984
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997
This theorem is referenced by: (None)
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