Theorem elnev 17489

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 2571	. . . . 5 |- _V = {x | x = x}
2	1	eqeq2i 2180	. . . 4 |- ({x | -. x = A} = _V <-> {x | -. x = A} = {x | x = x})
3		eq2ab 2282	. . . . 5 |- ({x | -. x = A} = {x | x = x} <-> A.x(-. x = A <-> x = x))
4		equid 1795	. . . . . . 7 |- x = x
5	4	tbt 409	. . . . . 6 |- (-. x = A <-> (-. x = A <-> x = x))
6	5	albii 1664	. . . . 5 |- (A.x -. x = A <-> A.x(-. x = A <-> x = x))
7	3, 6	bitr4i 310	. . . 4 |- ({x | -. x = A} = {x | x = x} <-> A.x -. x = A)
8		alnex 1698	. . . 4 |- (A.x -. x = A <-> -. E.x x = A)
9	2, 7, 8	3bitri 334	. . 3 |- ({x | -. x = A} = _V <-> -. E.x x = A)
10	9	con2bii 395	. 2 |- (E.x x = A <-> -. {x | -. x = A} = _V)
11		isset 2573	. 2 |- (A e. _V <-> E.x x = A)
12		df-ne 2297	. 2 |- ({x | -. x = A} =/= _V <-> -. {x | -. x = A} = _V)
13	10, 11, 12	3bitr4i 340	1 |- (A e. _V <-> {x | -. x = A} =/= _V)

Syntax hints:  -. wn 2   <-> wb 231  A.wal 1613   = wceq 1615   e. wcel 1617  E.wex 1644  {cab 2157   =/= wne 2295  _Vcvv 2569

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-10 1625  ax-12 1627  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893  ax-ext 2152

This theorem depends on definitions:  df-bi 232  df-an 435  df-ex 1645  df-sb 1845  df-clab 2158  df-cleq 2163  df-clel 2166  df-ne 2297  df-v 2571

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