Theorem eldifbd 3153

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3150. (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 3150	. . 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 2158   ∖ cdif 3138

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143

This theorem is referenced by:  fidifsnen  6884  fiunsnnn  6895  fimax2gtri  6915  unfidisj  6935  ssfirab  6947  fnfi  6950  iunfidisj  6959  hashunlem  10798  hashxp  10820  zfz1isolemiso  10833  fsumconst  11476  fsumrelem  11493  fprodcl2lem  11627  fprodconst  11642  fprodap0  11643  fprodrec  11651  fprodap0f  11658  fprodle  11662  fprodmodd  11663  fsumcncntop  14409  bj-charfun  14912  bj-charfundc  14913