Theorem dtru 1889

Description: A 'distinctor' is true in set theory. This says that at least two sets exist (or in terms of first-order logic, that the universe of discourse has two or more objects).

Assertion

Ref	Expression
dtru	|- -. A.x x = y

Distinct variable group(s):   x,y

Proof of Theorem dtru

Step	Hyp	Ref	Expression
1		0inp0 1888	. . . 4 |- (y = (/) -> -. y = {(/)})
2		p0ex 1885	. . . . 5 |- {(/)} e. V
3		cleq2 1110	. . . . . 6 |- (x = {(/)} -> (y = x <-> y = {(/)}))
4	3	negbid 463	. . . . 5 |- (x = {(/)} -> (-. y = x <-> -. y = {(/)}))
5	2, 4	cla4ev 1401	. . . 4 |- (-. y = {(/)} -> E.x -. y = x)
6	1, 5	syl 12	. . 3 |- (y = (/) -> E.x -. y = x)
7		0ex 1745	. . . 4 |- (/) e. V
8		cleq2 1110	. . . . 5 |- (x = (/) -> (y = x <-> y = (/)))
9	8	negbid 463	. . . 4 |- (x = (/) -> (-. y = x <-> -. y = (/)))
10	7, 9	cla4ev 1401	. . 3 |- (-. y = (/) -> E.x -. y = x)
11	6, 10	pm2.61i 110	. 2 |- E.x -. y = x
12		exnal 721	. . 3 |- (E.x -. y = x <-> -. A.x y = x)
13		cleqcom 1103	. . . . 5 |- (y = x <-> x = y)
14	13	bial 695	. . . 4 |- (A.x y = x <-> A.x x = y)
15	14	negbii 162	. . 3 |- (-. A.x y = x <-> -. A.x x = y)
16	12, 15	bitr 151	. 2 |- (E.x -. y = x <-> -. A.x x = y)
17	11, 16	mpbi 164	1 |- -. A.x x = y

Syntax hints:  -. wn 1  A.wal 672  E.wex 678   = weq 797   = wceq 1091  (/)c0 1707  {csn 1808

This theorem is referenced by:  dtrucor 1890  zfcndpow 3762

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812

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