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

Theorem undif3ss 3305
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 3185 . . . 4 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
2 eldif 3048 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
32orbi2i 734 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
4 orc 684 . . . . . . 7 (𝑥𝐴 → (𝑥𝐴𝑥𝐵))
5 olc 683 . . . . . . 7 (𝑥𝐴 → (¬ 𝑥𝐶𝑥𝐴))
64, 5jca 302 . . . . . 6 (𝑥𝐴 → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
7 olc 683 . . . . . . 7 (𝑥𝐵 → (𝑥𝐴𝑥𝐵))
8 orc 684 . . . . . . 7 𝑥𝐶 → (¬ 𝑥𝐶𝑥𝐴))
97, 8anim12i 334 . . . . . 6 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
106, 9jaoi 688 . . . . 5 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) → ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
11 simpl 108 . . . . . . 7 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → 𝑥𝐴)
1211orcd 705 . . . . . 6 ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
13 olc 683 . . . . . 6 ((𝑥𝐵 ∧ ¬ 𝑥𝐶) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
14 orc 684 . . . . . . 7 (𝑥𝐴 → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1514adantr 272 . . . . . 6 ((𝑥𝐴𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1614adantl 273 . . . . . 6 ((𝑥𝐵𝑥𝐴) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1712, 13, 15, 16ccase 931 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → (𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
1810, 17impbii 125 . . . 4 ((𝑥𝐴 ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
191, 3, 183bitri 205 . . 3 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)))
20 elun 3185 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2120biimpri 132 . . . . 5 ((𝑥𝐴𝑥𝐵) → 𝑥 ∈ (𝐴𝐵))
22 pm4.53r 723 . . . . . 6 ((¬ 𝑥𝐶𝑥𝐴) → ¬ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
23 eldif 3048 . . . . . 6 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐴))
2422, 23sylnibr 649 . . . . 5 ((¬ 𝑥𝐶𝑥𝐴) → ¬ 𝑥 ∈ (𝐶𝐴))
2521, 24anim12i 334 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
26 eldif 3048 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ ¬ 𝑥 ∈ (𝐶𝐴)))
2725, 26sylibr 133 . . 3 (((𝑥𝐴𝑥𝐵) ∧ (¬ 𝑥𝐶𝑥𝐴)) → 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
2819, 27sylbi 120 . 2 (𝑥 ∈ (𝐴 ∪ (𝐵𝐶)) → 𝑥 ∈ ((𝐴𝐵) ∖ (𝐶𝐴)))
2928ssriv 3069 1 (𝐴 ∪ (𝐵𝐶)) ⊆ ((𝐴𝐵) ∖ (𝐶𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wo 680  wcel 1463  cdif 3036  cun 3037  wss 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator