Theorem eldifbd 3133

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3130. (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 3130	. . 3 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
3	1, 2	sylib 121	. 2 |- ( ph -> ( A e. B /\ -. A e. C ) )
4	3	simprd 113	1 |- ( ph -> -. A e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 2141   \ cdif 3118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123

This theorem is referenced by:  fidifsnen  6848  fiunsnnn  6859  fimax2gtri  6879  unfidisj  6899  ssfirab  6911  fnfi  6914  iunfidisj  6923  hashunlem  10739  hashxp  10761  zfz1isolemiso  10774  fsumconst  11417  fsumrelem  11434  fprodcl2lem  11568  fprodconst  11583  fprodap0  11584  fprodrec  11592  fprodap0f  11599  fprodle  11603  fprodmodd  11604  fsumcncntop  13350  bj-charfun  13842  bj-charfundc  13843