Theorem ipo0 38653

Description: If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ipo0	|- ( _I Po A <-> A = (/) )

Proof of Theorem ipo0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equid 1939	. . . . 5 |- x = x
2		vex 3203	. . . . . 6 |- x e. _V
3	2	ideq 5274	. . . . 5 |- ( x _I x <-> x = x )
4	1, 3	mpbir 221	. . . 4 |- x _I x
5		poirr 5046	. . . . 5 |- ( ( _I Po A /\ x e. A ) -> -. x _I x )
6	5	ex 450	. . . 4 |- ( _I Po A -> ( x e. A -> -. x _I x ) )
7	4, 6	mt2i 132	. . 3 |- ( _I Po A -> -. x e. A )
8	7	eq0rdv 3979	. 2 |- ( _I Po A -> A = (/) )
9		po0 5050	. . 3 |- _I Po (/)
10		poeq2 5039	. . 3 |- ( A = (/) -> ( _I Po A <-> _I Po (/) ) )
11	9, 10	mpbiri 248	. 2 |- ( A = (/) -> _I Po A )
12	8, 11	impbii 199	1 |- ( _I Po A <-> A = (/) )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. wcel 1990   (/)c0 3915   class class class wbr 4653   _I cid 5023   Po wpo 5033

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-po 5035  df-xp 5120  df-rel 5121

This theorem is referenced by: (None)