Theorem intasym 5511

Description: Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
intasym	|- ( ( R i^i `' R ) C_ _I <-> A. x A. y ( ( x R y /\ y R x ) -> x = y ) )

Distinct variable group:   x, y, R

Proof of Theorem intasym

Step	Hyp	Ref	Expression
1		relcnv 5503	. . 3 |- Rel `' R
2		relin2 5237	. . 3 |- ( Rel `' R -> Rel ( R i^i `' R ) )
3		ssrel 5207	. . 3 |- ( Rel ( R i^i `' R ) -> ( ( R i^i `' R ) C_ _I <-> A. x A. y ( <. x , y >. e. ( R i^i `' R ) -> <. x , y >. e. _I ) ) )
4	1, 2, 3	mp2b 10	. 2 |- ( ( R i^i `' R ) C_ _I <-> A. x A. y ( <. x , y >. e. ( R i^i `' R ) -> <. x , y >. e. _I ) )
5		elin 3796	. . . . 5 |- ( <. x , y >. e. ( R i^i `' R ) <-> ( <. x , y >. e. R /\ <. x , y >. e. `' R ) )
6		df-br 4654	. . . . . 6 |- ( x R y <-> <. x , y >. e. R )
7		vex 3203	. . . . . . . 8 |- x e. _V
8		vex 3203	. . . . . . . 8 |- y e. _V
9	7, 8	brcnv 5305	. . . . . . 7 |- ( x `' R y <-> y R x )
10		df-br 4654	. . . . . . 7 |- ( x `' R y <-> <. x , y >. e. `' R )
11	9, 10	bitr3i 266	. . . . . 6 |- ( y R x <-> <. x , y >. e. `' R )
12	6, 11	anbi12i 733	. . . . 5 |- ( ( x R y /\ y R x ) <-> ( <. x , y >. e. R /\ <. x , y >. e. `' R ) )
13	5, 12	bitr4i 267	. . . 4 |- ( <. x , y >. e. ( R i^i `' R ) <-> ( x R y /\ y R x ) )
14		df-br 4654	. . . . 5 |- ( x _I y <-> <. x , y >. e. _I )
15	8	ideq 5274	. . . . 5 |- ( x _I y <-> x = y )
16	14, 15	bitr3i 266	. . . 4 |- ( <. x , y >. e. _I <-> x = y )
17	13, 16	imbi12i 340	. . 3 |- ( ( <. x , y >. e. ( R i^i `' R ) -> <. x , y >. e. _I ) <-> ( ( x R y /\ y R x ) -> x = y ) )
18	17	2albii 1748	. 2 |- ( A. x A. y ( <. x , y >. e. ( R i^i `' R ) -> <. x , y >. e. _I ) <-> A. x A. y ( ( x R y /\ y R x ) -> x = y ) )
19	4, 18	bitri 264	1 |- ( ( R i^i `' R ) C_ _I <-> A. x A. y ( ( x R y /\ y R x ) -> x = y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   i^i cin 3573   C_ wss 3574   <.cop 4183   class class class wbr 4653   _I cid 5023   `'ccnv 5113   Rel wrel 5119

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-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by: (None)