Theorem neldifsn 3841

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

Assertion

Ref	Expression
neldifsn	⊢ ¬ A ∈ (B ∖ {A})

Proof of Theorem neldifsn

Step	Hyp	Ref	Expression
1		neirr 2521	. 2 ⊢ ¬ A ≠ A
2		eldifsni 3840	. 2 ⊢ (A ∈ (B ∖ {A}) → A ≠ A)
3	1, 2	mto 167	1 ⊢ ¬ A ∈ (B ∖ {A})

Syntax hints:  ¬ wn 3   ∈ wcel 1710   ≠ wne 2516   ∖ cdif 3206  {csn 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-sn 3741

This theorem is referenced by:  neldifsnd  3842  phiall  4618