Theorem eldifad 3211

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3209. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
eldifad.1	|- ( ph -> A e. ( B \ C ) )

Assertion

Ref	Expression
eldifad	|- ( ph -> A e. B )

Proof of Theorem eldifad

Step	Hyp	Ref	Expression
1		eldifad.1	. . 3 |- ( ph -> A e. ( B \ C ) )
2		eldif 3209	. . 3 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
3	1, 2	sylib 122	. 2 |- ( ph -> ( A e. B /\ -. A e. C ) )
4	3	simpld 112	1 |- ( ph -> A e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2202   \ cdif 3197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202

This theorem is referenced by:  fimax2gtri  7091  finexdc  7092  elssdc  7094  unfidisj  7114  undifdc  7116  ssfirab  7129  fnfi  7135  iunfidisj  7145  dcfi  7180  hashunlem  11067  zfz1isolemiso  11103  fsumrelem  12033  fprodcl2lem  12167  fprodap0  12183  fprodrec  12191  fprodap0f  12198  fprodle  12202  iuncld  14841  fsumcncntop  15293  gausslemma2dlem0i  15788  gausslemma2dlem4  15795  gausslemma2dlem5a  15796  gausslemma2dlem7  15799  lgseisenlem1  15801  lgseisenlem2  15802  lgseisenlem3  15803  lgseisenlem4  15804  lgseisen  15805  lgsquadlem1  15808  lgsquadlem2  15809  lgsquadlem3  15810  1loopgrvd0fi  16159  bj-charfun  16405  gfsumcl  16690