Theorem eldifad 3208

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3206. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
eldifad.1	⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))

Assertion

Ref	Expression
eldifad	⊢ (𝜑 → 𝐴 ∈ 𝐵)

Proof of Theorem eldifad

Step	Hyp	Ref	Expression
1		eldifad.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))
2		eldif 3206	. . 3 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
4	3	simpld 112	1 ⊢ (𝜑 → 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∈ wcel 2200   ∖ cdif 3194

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199

This theorem is referenced by:  fimax2gtri  7072  finexdc  7073  elssdc  7075  unfidisj  7095  undifdc  7097  ssfirab  7109  fnfi  7114  iunfidisj  7124  dcfi  7159  hashunlem  11038  zfz1isolemiso  11074  fsumrelem  11997  fprodcl2lem  12131  fprodap0  12147  fprodrec  12155  fprodap0f  12162  fprodle  12166  iuncld  14804  fsumcncntop  15256  gausslemma2dlem0i  15751  gausslemma2dlem4  15758  gausslemma2dlem5a  15759  gausslemma2dlem7  15762  lgseisenlem1  15764  lgseisenlem2  15765  lgseisenlem3  15766  lgseisenlem4  15767  lgseisen  15768  lgsquadlem1  15771  lgsquadlem2  15772  lgsquadlem3  15773  bj-charfun  16225