Theorem intirr 5514

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

Assertion

Ref	Expression
intirr	|- ( ( R i^i _I ) = (/) <-> A. x -. x R x )

Distinct variable group:   x, R

Proof of Theorem intirr

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		incom 3805	. . . 4 |- ( R i^i _I ) = ( _I i^i R )
2	1	eqeq1i 2627	. . 3 |- ( ( R i^i _I ) = (/) <-> ( _I i^i R ) = (/) )
3		disj2 4024	. . 3 |- ( ( _I i^i R ) = (/) <-> _I C_ ( _V \ R ) )
4		reli 5249	. . . 4 |- Rel _I
5		ssrel 5207	. . . 4 |- ( Rel _I -> ( _I C_ ( _V \ R ) <-> A. x A. y ( <. x , y >. e. _I -> <. x , y >. e. ( _V \ R ) ) ) )
6	4, 5	ax-mp 5	. . 3 |- ( _I C_ ( _V \ R ) <-> A. x A. y ( <. x , y >. e. _I -> <. x , y >. e. ( _V \ R ) ) )
7	2, 3, 6	3bitri 286	. 2 |- ( ( R i^i _I ) = (/) <-> A. x A. y ( <. x , y >. e. _I -> <. x , y >. e. ( _V \ R ) ) )
8		equcom 1945	. . . . 5 |- ( y = x <-> x = y )
9		vex 3203	. . . . . 6 |- y e. _V
10	9	ideq 5274	. . . . 5 |- ( x _I y <-> x = y )
11		df-br 4654	. . . . 5 |- ( x _I y <-> <. x , y >. e. _I )
12	8, 10, 11	3bitr2i 288	. . . 4 |- ( y = x <-> <. x , y >. e. _I )
13		opex 4932	. . . . . . 7 |- <. x , y >. e. _V
14	13	biantrur 527	. . . . . 6 |- ( -. <. x , y >. e. R <-> ( <. x , y >. e. _V /\ -. <. x , y >. e. R ) )
15		eldif 3584	. . . . . 6 |- ( <. x , y >. e. ( _V \ R ) <-> ( <. x , y >. e. _V /\ -. <. x , y >. e. R ) )
16	14, 15	bitr4i 267	. . . . 5 |- ( -. <. x , y >. e. R <-> <. x , y >. e. ( _V \ R ) )
17		df-br 4654	. . . . 5 |- ( x R y <-> <. x , y >. e. R )
18	16, 17	xchnxbir 323	. . . 4 |- ( -. x R y <-> <. x , y >. e. ( _V \ R ) )
19	12, 18	imbi12i 340	. . 3 |- ( ( y = x -> -. x R y ) <-> ( <. x , y >. e. _I -> <. x , y >. e. ( _V \ R ) ) )
20	19	2albii 1748	. 2 |- ( A. x A. y ( y = x -> -. x R y ) <-> A. x A. y ( <. x , y >. e. _I -> <. x , y >. e. ( _V \ R ) ) )
21		breq2 4657	. . . . 5 |- ( y = x -> ( x R y <-> x R x ) )
22	21	notbid 308	. . . 4 |- ( y = x -> ( -. x R y <-> -. x R x ) )
23	22	equsalvw 1931	. . 3 |- ( A. y ( y = x -> -. x R y ) <-> -. x R x )
24	23	albii 1747	. 2 |- ( A. x A. y ( y = x -> -. x R y ) <-> A. x -. x R x )
25	7, 20, 24	3bitr2i 288	1 |- ( ( R i^i _I ) = (/) <-> A. x -. x R x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   <.cop 4183   class class class wbr 4653   _I cid 5023   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

This theorem is referenced by:  hartogslem1  8447  hausdiag  21448