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Theorem undif3ss 3171
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 3057 . . . 4 (x (A ∪ (B𝐶)) ↔ (x A x (B𝐶)))
2 eldif 2900 . . . . 5 (x (B𝐶) ↔ (x B ¬ x 𝐶))
32orbi2i 666 . . . 4 ((x A x (B𝐶)) ↔ (x A (x B ¬ x 𝐶)))
4 orc 620 . . . . . . 7 (x A → (x A x B))
5 olc 619 . . . . . . 7 (x A → (¬ x 𝐶 x A))
64, 5jca 290 . . . . . 6 (x A → ((x A x B) x 𝐶 x A)))
7 olc 619 . . . . . . 7 (x B → (x A x B))
8 orc 620 . . . . . . 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 623 . . . . 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 639 . . . . . 6 ((x A ¬ x 𝐶) → (x A (x B ¬ x 𝐶)))
13 olc 619 . . . . . 6 ((x B ¬ x 𝐶) → (x A (x B ¬ x 𝐶)))
14 orc 620 . . . . . . 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 857 . . . . 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 3057 . . . . . 6 (x (AB) ↔ (x A x B))
2120biimpri 124 . . . . 5 ((x A x B) → x (AB))
22 pm4.53r 792 . . . . . 6 ((¬ x 𝐶 x A) → ¬ (x 𝐶 ¬ x A))
23 eldif 2900 . . . . . 6 (x (𝐶A) ↔ (x 𝐶 ¬ x A))
2422, 23sylnibr 589 . . . . 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 2900 . . . 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 2922 1 (A ∪ (B𝐶)) ⊆ ((AB) ∖ (𝐶A))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 616   wcel 1370  cdif 2887  cun 2888  wss 2890
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904
This theorem is referenced by: (None)
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