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Theorem ssdisj 3271
 Description: Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
Assertion
Ref Expression
ssdisj ((AB (B𝐶) = ∅) → (A𝐶) = ∅)

Proof of Theorem ssdisj
StepHypRef Expression
1 ss0b 3250 . . . 4 ((B𝐶) ⊆ ∅ ↔ (B𝐶) = ∅)
2 ssrin 3156 . . . . 5 (AB → (A𝐶) ⊆ (B𝐶))
3 sstr2 2946 . . . . 5 ((A𝐶) ⊆ (B𝐶) → ((B𝐶) ⊆ ∅ → (A𝐶) ⊆ ∅))
42, 3syl 14 . . . 4 (AB → ((B𝐶) ⊆ ∅ → (A𝐶) ⊆ ∅))
51, 4syl5bir 142 . . 3 (AB → ((B𝐶) = ∅ → (A𝐶) ⊆ ∅))
65imp 115 . 2 ((AB (B𝐶) = ∅) → (A𝐶) ⊆ ∅)
7 ss0 3251 . 2 ((A𝐶) ⊆ ∅ → (A𝐶) = ∅)
86, 7syl 14 1 ((AB (B𝐶) = ∅) → (A𝐶) = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∩ 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-in 2918  df-ss 2925  df-nul 3219 This theorem is referenced by:  djudisj  4693  fimacnvdisj  5017
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