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Theorem disjssun 3262
 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 3161 . . . . 5 (A ∩ (B𝐶)) = ((AB) ∪ (A𝐶))
21equncomi 3066 . . . 4 (A ∩ (B𝐶)) = ((A𝐶) ∪ (AB))
3 uneq2 3068 . . . . 5 ((AB) = ∅ → ((A𝐶) ∪ (AB)) = ((A𝐶) ∪ ∅))
4 un0 3228 . . . . 5 ((A𝐶) ∪ ∅) = (A𝐶)
53, 4syl6eq 2070 . . . 4 ((AB) = ∅ → ((A𝐶) ∪ (AB)) = (A𝐶))
62, 5syl5eq 2066 . . 3 ((AB) = ∅ → (A ∩ (B𝐶)) = (A𝐶))
76eqeq1d 2030 . 2 ((AB) = ∅ → ((A ∩ (B𝐶)) = A ↔ (A𝐶) = A))
8 df-ss 2908 . 2 (A ⊆ (B𝐶) ↔ (A ∩ (B𝐶)) = A)
9 df-ss 2908 . 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 1228   ∪ cun 2892   ∩ cin 2893   ⊆ wss 2894  ∅c0 3201 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202 This theorem is referenced by: (None)
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