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Theorem ssdifin0 3631
 Description: A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
ssdifin0 (A (B C) → (AC) = )

Proof of Theorem ssdifin0
StepHypRef Expression
1 ssrin 3480 . 2 (A (B C) → (AC) ((B C) ∩ C))
2 incom 3448 . . 3 ((B C) ∩ C) = (C ∩ (B C))
3 disjdif 3622 . . 3 (C ∩ (B C)) =
42, 3eqtri 2373 . 2 ((B C) ∩ C) =
5 sseq0 3582 . 2 (((AC) ((B C) ∩ C) ((B C) ∩ C) = ) → (AC) = )
61, 4, 5sylancl 643 1 (A (B C) → (AC) = )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∖ cdif 3206   ∩ 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-ne 2518  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:  ssdifeq0  3632
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