Theorem eldifvsn 3810

Description: A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019.)

Assertion

Ref	Expression
eldifvsn	|- ( A e. V -> ( A e. ( _V \ { B } ) <-> A =/= B ) )

Proof of Theorem eldifvsn

Step	Hyp	Ref	Expression
1		eldifsn 3804	. 2 |- ( A e. ( _V \ { B } ) <-> ( A e. _V /\ A =/= B ) )
2		elex 2815	. . 3 |- ( A e. V -> A e. _V )
3	2	biantrurd 305	. 2 |- ( A e. V -> ( A =/= B <-> ( A e. _V /\ A =/= B ) ) )
4	1, 3	bitr4id 199	1 |- ( A e. V -> ( A e. ( _V \ { B } ) <-> A =/= B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2202   =/= wne 2403   _Vcvv 2803   \ cdif 3198   {csn 3673

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-dif 3203  df-sn 3679

This theorem is referenced by:  cnvimadfsn  6423