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  11068  zfz1isolemiso  11104  fsumrelem  12034  fprodcl2lem  12168  fprodap0  12184  fprodrec  12192  fprodap0f  12199  fprodle  12203  iuncld  14842  fsumcncntop  15294  gausslemma2dlem0i  15789  gausslemma2dlem4  15796  gausslemma2dlem5a  15797  gausslemma2dlem7  15800  lgseisenlem1  15802  lgseisenlem2  15803  lgseisenlem3  15804  lgseisenlem4  15805  lgseisen  15806  lgsquadlem1  15809  lgsquadlem2  15810  lgsquadlem3  15811  1loopgrvd0fi  16160  bj-charfun  16419  gfsumcl  16704