Theorem elnev 16404

Description: Any set that contains one element less than the universe is not equal to it.

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		df-v 2294	. . . . 5 |- _V = {x | x = x}
2	1	eqeq2i 1894	. . . 4 |- ({x | -. x = A} = _V <-> {x | -. x = A} = {x | x = x})
3		eq2ab 2004	. . . . 5 |- ({x | -. x = A} = {x | x = x} <-> A.x(-. x = A <-> x = x))
4		equid 1484	. . . . . . 7 |- x = x
5	4	tbt 788	. . . . . 6 |- (-. x = A <-> (-. x = A <-> x = x))
6	5	albii 1346	. . . . 5 |- (A.x -. x = A <-> A.x(-. x = A <-> x = x))
7	3, 6	bitr4i 193	. . . 4 |- ({x | -. x = A} = {x | x = x} <-> A.x -. x = A)
8		alnex 1380	. . . 4 |- (A.x -. x = A <-> -. E.x x = A)
9	2, 7, 8	3bitri 194	. . 3 |- ({x | -. x = A} = _V <-> -. E.x x = A)
10	9	con2bii 238	. 2 |- (E.x x = A <-> -. {x | -. x = A} = _V)
11		isset 2296	. 2 |- (A e. _V <-> E.x x = A)
12		df-ne 2019	. 2 |- ({x | -. x = A} =/= _V <-> -. {x | -. x = A} = _V)
13	10, 11, 12	3bitr4i 200	1 |- (A e. _V <-> {x | -. x = A} =/= _V)

Syntax hints:  -. wn 2   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  _Vcvv 2292

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294

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