Theorem eldifad 3185

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3183. (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 3183	. . 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 2178   ∖ cdif 3171

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176

This theorem is referenced by:  fimax2gtri  7024  finexdc  7025  unfidisj  7045  undifdc  7047  ssfirab  7059  fnfi  7064  iunfidisj  7074  dcfi  7109  hashunlem  10986  zfz1isolemiso  11021  fsumrelem  11897  fprodcl2lem  12031  fprodap0  12047  fprodrec  12055  fprodap0f  12062  fprodle  12066  iuncld  14702  fsumcncntop  15154  gausslemma2dlem0i  15649  gausslemma2dlem4  15656  gausslemma2dlem5a  15657  gausslemma2dlem7  15660  lgseisenlem1  15662  lgseisenlem2  15663  lgseisenlem3  15664  lgseisenlem4  15665  lgseisen  15666  lgsquadlem1  15669  lgsquadlem2  15670  lgsquadlem3  15671  bj-charfun  15942