Theorem eldifvsn 3845

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 3839	. 2 |- ( A e. ( _V \ { B } ) <-> ( A e. _V /\ A =/= B ) )
2		elex 2833	. . 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 2209   =/= wne 2420   _Vcvv 2821   \ cdif 3217   {csn 3708

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-sn 3714

This theorem is referenced by:  cnvimadfsn  6479