Theorem eldifad 3164

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3162. (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 3162	. . 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 2164   ∖ cdif 3150

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155

This theorem is referenced by:  fimax2gtri  6957  finexdc  6958  unfidisj  6978  undifdc  6980  ssfirab  6990  fnfi  6995  iunfidisj  7005  dcfi  7040  hashunlem  10875  zfz1isolemiso  10910  fsumrelem  11614  fprodcl2lem  11748  fprodap0  11764  fprodrec  11772  fprodap0f  11779  fprodle  11783  iuncld  14283  fsumcncntop  14724  gausslemma2dlem0i  15173  gausslemma2dlem4  15180  gausslemma2dlem5a  15181  gausslemma2dlem7  15184  lgseisenlem1  15186  lgseisenlem2  15187  lgseisenlem3  15188  lgseisenlem4  15189  lgseisen  15190  lgsquadlem1  15191  bj-charfun  15299