Theorem eldifbd 3223

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3220. (Contributed by David Moews, 1-May-2017.)

Hypothesis

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

Assertion

Ref	Expression
eldifbd	⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)

Proof of Theorem eldifbd

Step	Hyp	Ref	Expression
1		eldifbd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶))
2		eldif 3220	. . 3 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
4	3	simprd 114	1 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∈ wcel 2203   ∖ cdif 3208

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213

This theorem is referenced by:  fvdifsuppst  6444  fidifsnen  7125  fiunsnnn  7138  fimax2gtri  7159  unfidisj  7182  ssfirab  7197  fnfi  7203  iunfidisj  7213  mapfi  7214  hashunlem  11168  hashxp  11191  zfz1isolemiso  11211  fsumconst  12140  fsumrelem  12157  fprodcl2lem  12291  fprodconst  12306  fprodap0  12307  fprodrec  12315  fprodap0f  12322  fprodle  12326  fprodmodd  12327  fsumcncntop  15432  1loopgrvd0fi  16301  bj-charfun  16577  bj-charfundc  16578  gfsumz  16869  gfsumcl  16870