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Theorem disjssun 3279
Description: Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
disjssun ((AB) = ∅ → (A ⊆ (B𝐶) ↔ A𝐶))

Proof of Theorem disjssun
StepHypRef Expression
1 indi 3178 . . . . 5 (A ∩ (B𝐶)) = ((AB) ∪ (A𝐶))
21equncomi 3083 . . . 4 (A ∩ (B𝐶)) = ((A𝐶) ∪ (AB))
3 uneq2 3085 . . . . 5 ((AB) = ∅ → ((A𝐶) ∪ (AB)) = ((A𝐶) ∪ ∅))
4 un0 3245 . . . . 5 ((A𝐶) ∪ ∅) = (A𝐶)
53, 4syl6eq 2085 . . . 4 ((AB) = ∅ → ((A𝐶) ∪ (AB)) = (A𝐶))
62, 5syl5eq 2081 . . 3 ((AB) = ∅ → (A ∩ (B𝐶)) = (A𝐶))
76eqeq1d 2045 . 2 ((AB) = ∅ → ((A ∩ (B𝐶)) = A ↔ (A𝐶) = A))
8 df-ss 2925 . 2 (A ⊆ (B𝐶) ↔ (A ∩ (B𝐶)) = A)
9 df-ss 2925 . 2 (A𝐶 ↔ (A𝐶) = A)
107, 8, 93bitr4g 212 1 ((AB) = ∅ → (A ⊆ (B𝐶) ↔ A𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  cun 2909  cin 2910  wss 2911  c0 3218
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  df-nul 3219
This theorem is referenced by: (None)
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