Theorem neldifsnd 3749

Description: 𝐴 is not in (𝐵 ∖ {𝐴}). Deduction form. (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
neldifsnd	⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))

Proof of Theorem neldifsnd

Step	Hyp	Ref	Expression
1		neldifsn 3748	. 2 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
2	1	a1i 9	1 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2164   ∖ cdif 3150  {csn 3618

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-ne 2365  df-v 2762  df-dif 3155  df-sn 3624

This theorem is referenced by:  difsnb  3761  frirrg  4381  elirr  4573