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Theorem ssdisj 3600
 Description: Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
Assertion
Ref Expression
ssdisj ((A B (BC) = ) → (AC) = )

Proof of Theorem ssdisj
StepHypRef Expression
1 ss0b 3580 . . . 4 ((BC) ↔ (BC) = )
2 ssrin 3480 . . . . 5 (A B → (AC) (BC))
3 sstr2 3279 . . . . 5 ((AC) (BC) → ((BC) → (AC) ))
42, 3syl 15 . . . 4 (A B → ((BC) → (AC) ))
51, 4syl5bir 209 . . 3 (A B → ((BC) = → (AC) ))
65imp 418 . 2 ((A B (BC) = ) → (AC) )
7 ss0 3581 . 2 ((AC) → (AC) = )
86, 7syl 15 1 ((A B (BC) = ) → (AC) = )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by:  fimacnvdisj  5244
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