Theorem eldifad 3168

Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3166. (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 3166	. . 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 2167   ∖ cdif 3154

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159

This theorem is referenced by:  fimax2gtri  6964  finexdc  6965  unfidisj  6985  undifdc  6987  ssfirab  6999  fnfi  7004  iunfidisj  7014  dcfi  7049  hashunlem  10899  zfz1isolemiso  10934  fsumrelem  11639  fprodcl2lem  11773  fprodap0  11789  fprodrec  11797  fprodap0f  11804  fprodle  11808  iuncld  14377  fsumcncntop  14829  gausslemma2dlem0i  15324  gausslemma2dlem4  15331  gausslemma2dlem5a  15332  gausslemma2dlem7  15335  lgseisenlem1  15337  lgseisenlem2  15338  lgseisenlem3  15339  lgseisenlem4  15340  lgseisen  15341  lgsquadlem1  15344  lgsquadlem2  15345  lgsquadlem3  15346  bj-charfun  15479