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Theorem undif3ss 3225
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 3111 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
2 eldif 2954 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
32orbi2i 689 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4 orc 643 . . . . . . 7 (𝑥𝐴 → (𝑥𝐴𝑥𝐵))
5 olc 642 . . . . . . 7 (𝑥𝐴 → (¬ 𝑥𝐶𝑥𝐴))
64, 5jca 294 . . . . . 6 (𝑥𝐴 → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
7 olc 642 . . . . . . 7 (𝑥𝐵 → (𝑥𝐴𝑥𝐵))
8 orc 643 . . . . . . 7 𝑥𝐶 → (¬ 𝑥𝐶𝑥𝐴))
97, 8anim12i 325 . . . . . 6 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
106, 9jaoi 646 . . . . 5 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
11 simpl 106 . . . . . . 7 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → 𝑥𝐴)
1211orcd 662 . . . . . 6 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
13 olc 642 . . . . . 6 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
14 orc 643 . . . . . . 7 (𝑥𝐴 → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1514adantr 265 . . . . . 6 ((𝑥𝐴𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1614adantl 266 . . . . . 6 ((𝑥𝐵𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1712, 13, 15, 16ccase 882 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1810, 17impbii 121 . . . 4 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
191, 3, 183bitri 199 . . 3 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
20 elun 3111 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2120biimpri 128 . . . . 5 ((𝑥𝐴𝑥𝐵) → 𝑥 ∈ (𝐴𝐵))
22 pm4.53r 815 . . . . . 6 ((¬ 𝑥𝐶𝑥𝐴) → ¬ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
23 eldif 2954 . . . . . 6 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
2422, 23sylnibr 612 . . . . 5 ((¬ 𝑥𝐶𝑥𝐴) → ¬ 𝑥 ∈ (𝐶𝐴))
2521, 24anim12i 325 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
26 eldif 2954 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
2725, 26sylibr 141 . . 3 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
2819, 27sylbi 118 . 2 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) → 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
2928ssriv 2976 1 (𝐴 ∪ (𝐵𝐶)) ⊆ ((𝐴𝐵) ∖ (𝐶𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101  wo 639  wcel 1409  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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