Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  undif3ss GIF version

Theorem undif3ss 3198
 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 3084 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
2 eldif 2927 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
32orbi2i 679 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4 orc 633 . . . . . . 7 (𝑥𝐴 → (𝑥𝐴𝑥𝐵))
5 olc 632 . . . . . . 7 (𝑥𝐴 → (¬ 𝑥𝐶𝑥𝐴))
64, 5jca 290 . . . . . 6 (𝑥𝐴 → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
7 olc 632 . . . . . . 7 (𝑥𝐵 → (𝑥𝐴𝑥𝐵))
8 orc 633 . . . . . . 7 𝑥𝐶 → (¬ 𝑥𝐶𝑥𝐴))
97, 8anim12i 321 . . . . . 6 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
106, 9jaoi 636 . . . . 5 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
11 simpl 102 . . . . . . 7 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → 𝑥𝐴)
1211orcd 652 . . . . . 6 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
13 olc 632 . . . . . 6 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
14 orc 633 . . . . . . 7 (𝑥𝐴 → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1514adantr 261 . . . . . 6 ((𝑥𝐴𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1614adantl 262 . . . . . 6 ((𝑥𝐵𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1712, 13, 15, 16ccase 871 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1810, 17impbii 117 . . . 4 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
191, 3, 183bitri 195 . . 3 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
20 elun 3084 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2120biimpri 124 . . . . 5 ((𝑥𝐴𝑥𝐵) → 𝑥 ∈ (𝐴𝐵))
22 pm4.53r 804 . . . . . 6 ((¬ 𝑥𝐶𝑥𝐴) → ¬ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
23 eldif 2927 . . . . . 6 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
2422, 23sylnibr 602 . . . . 5 ((¬ 𝑥𝐶𝑥𝐴) → ¬ 𝑥 ∈ (𝐶𝐴))
2521, 24anim12i 321 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
26 eldif 2927 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
2725, 26sylibr 137 . . 3 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
2819, 27sylbi 114 . 2 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) → 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
2928ssriv 2949 1 (𝐴 ∪ (𝐵𝐶)) ⊆ ((𝐴𝐵) ∖ (𝐶𝐴))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ∨ wo 629   ∈ wcel 1393   ∖ cdif 2914   ∪ cun 2915   ⊆ wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator