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Theorem undif3ss 3192
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 3078 . . . 4 (x (A ∪ (B𝐶)) ↔ (x A x (B𝐶)))
2 eldif 2921 . . . . 5 (x (B𝐶) ↔ (x B ¬ x 𝐶))
32orbi2i 678 . . . 4 ((x A x (B𝐶)) ↔ (x A (x B ¬ x 𝐶)))
4 orc 632 . . . . . . 7 (x A → (x A x B))
5 olc 631 . . . . . . 7 (x A → (¬ x 𝐶 x A))
64, 5jca 290 . . . . . 6 (x A → ((x A x B) x 𝐶 x A)))
7 olc 631 . . . . . . 7 (x B → (x A x B))
8 orc 632 . . . . . . 7 x 𝐶 → (¬ x 𝐶 x A))
97, 8anim12i 321 . . . . . 6 ((x B ¬ x 𝐶) → ((x A x B) x 𝐶 x A)))
106, 9jaoi 635 . . . . 5 ((x A (x B ¬ x 𝐶)) → ((x A x B) x 𝐶 x A)))
11 simpl 102 . . . . . . 7 ((x A ¬ x 𝐶) → x A)
1211orcd 651 . . . . . 6 ((x A ¬ x 𝐶) → (x A (x B ¬ x 𝐶)))
13 olc 631 . . . . . 6 ((x B ¬ x 𝐶) → (x A (x B ¬ x 𝐶)))
14 orc 632 . . . . . . 7 (x A → (x A (x B ¬ x 𝐶)))
1514adantr 261 . . . . . 6 ((x A x A) → (x A (x B ¬ x 𝐶)))
1614adantl 262 . . . . . 6 ((x B x A) → (x A (x B ¬ x 𝐶)))
1712, 13, 15, 16ccase 870 . . . . 5 (((x A x B) x 𝐶 x A)) → (x A (x B ¬ x 𝐶)))
1810, 17impbii 117 . . . 4 ((x A (x B ¬ x 𝐶)) ↔ ((x A x B) x 𝐶 x A)))
191, 3, 183bitri 195 . . 3 (x (A ∪ (B𝐶)) ↔ ((x A x B) x 𝐶 x A)))
20 elun 3078 . . . . . 6 (x (AB) ↔ (x A x B))
2120biimpri 124 . . . . 5 ((x A x B) → x (AB))
22 pm4.53r 803 . . . . . 6 ((¬ x 𝐶 x A) → ¬ (x 𝐶 ¬ x A))
23 eldif 2921 . . . . . 6 (x (𝐶A) ↔ (x 𝐶 ¬ x A))
2422, 23sylnibr 601 . . . . 5 ((¬ x 𝐶 x A) → ¬ x (𝐶A))
2521, 24anim12i 321 . . . 4 (((x A x B) x 𝐶 x A)) → (x (AB) ¬ x (𝐶A)))
26 eldif 2921 . . . 4 (x ((AB) ∖ (𝐶A)) ↔ (x (AB) ¬ x (𝐶A)))
2725, 26sylibr 137 . . 3 (((x A x B) x 𝐶 x A)) → x ((AB) ∖ (𝐶A)))
2819, 27sylbi 114 . 2 (x (A ∪ (B𝐶)) → x ((AB) ∖ (𝐶A)))
2928ssriv 2943 1 (A ∪ (B𝐶)) ⊆ ((AB) ∖ (𝐶A))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 628   wcel 1390  cdif 2908  cun 2909  wss 2911
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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