Theorem elnev 38639

Description: Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.)

Assertion

Ref	Expression
elnev	|- ( A e. _V <-> { x | -. x = A } =/= _V )

Distinct variable group:   x, A

Proof of Theorem elnev

Step	Hyp	Ref	Expression
1		isset 3207	. 2 |- ( A e. _V <-> E. x x = A )
2		df-v 3202	. . . . 5 |- _V = { x | x = x }
3	2	eqeq2i 2634	. . . 4 |- ( { x | -. x = A } = _V <-> { x | -. x = A } = { x | x = x } )
4		equid 1939	. . . . . . 7 |- x = x
5	4	tbt 359	. . . . . 6 |- ( -. x = A <-> ( -. x = A <-> x = x ) )
6	5	albii 1747	. . . . 5 |- ( A. x -. x = A <-> A. x ( -. x = A <-> x = x ) )
7		alnex 1706	. . . . 5 |- ( A. x -. x = A <-> -. E. x x = A )
8		abbi 2737	. . . . 5 |- ( A. x ( -. x = A <-> x = x ) <-> { x | -. x = A } = { x | x = x } )
9	6, 7, 8	3bitr3ri 291	. . . 4 |- ( { x | -. x = A } = { x | x = x } <-> -. E. x x = A )
10	3, 9	bitri 264	. . 3 |- ( { x | -. x = A } = _V <-> -. E. x x = A )
11	10	necon2abii 2844	. 2 |- ( E. x x = A <-> { x | -. x = A } =/= _V )
12	1, 11	bitri 264	1 |- ( A e. _V <-> { x | -. x = A } =/= _V )

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-v 3202

This theorem is referenced by: (None)