Theorem eldifbd 3225

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3222. (Contributed by David Moews, 1-May-2017.)

Hypothesis

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

Assertion

Ref	Expression
eldifbd	|- ( ph -> -. A e. C )

Proof of Theorem eldifbd

Step	Hyp	Ref	Expression
1		eldifbd.1	. . 3 |- ( ph -> A e. ( B \ C ) )
2		eldif 3222	. . 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	simprd 114	1 |- ( ph -> -. A e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2205   \ cdif 3210

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-dif 3215

This theorem is referenced by:  fvdifsuppst  6446  fidifsnen  7127  fiunsnnn  7140  fimax2gtri  7161  unfidisj  7184  ssfirab  7199  fnfi  7205  iunfidisj  7215  mapfi  7216  hashunlem  11172  hashxp  11195  zfz1isolemiso  11215  fsumconst  12144  fsumrelem  12161  fprodcl2lem  12295  fprodconst  12310  fprodap0  12311  fprodrec  12319  fprodap0f  12326  fprodle  12330  fprodmodd  12331  fsumcncntop  15449  1loopgrvd0fi  16318  bj-charfun  16594  bj-charfundc  16595  gfsumz  16886  gfsumcl  16887