Theorem opeldifid 29412

Description: Ordered pair elementhood outside of the diagonal. (Contributed by Thierry Arnoux, 1-Jan-2020.)

Assertion

Ref	Expression
opeldifid	|- ( Rel A -> ( <. X , Y >. e. ( A \ _I ) <-> ( <. X , Y >. e. A /\ X =/= Y ) ) )

Proof of Theorem opeldifid

Step	Hyp	Ref	Expression
1		reldif 5238	. . . 4 |- ( Rel A -> Rel ( A \ _I ) )
2		brrelex2 5157	. . . 4 |- ( ( Rel ( A \ _I ) /\ X ( A \ _I ) Y ) -> Y e. _V )
3	1, 2	sylan 488	. . 3 |- ( ( Rel A /\ X ( A \ _I ) Y ) -> Y e. _V )
4		brrelex2 5157	. . . 4 |- ( ( Rel A /\ X A Y ) -> Y e. _V )
5	4	adantrr 753	. . 3 |- ( ( Rel A /\ ( X A Y /\ X =/= Y ) ) -> Y e. _V )
6		brdif 4705	. . . 4 |- ( X ( A \ _I ) Y <-> ( X A Y /\ -. X _I Y ) )
7		ideqg 5273	. . . . . 6 |- ( Y e. _V -> ( X _I Y <-> X = Y ) )
8	7	necon3bbid 2831	. . . . 5 |- ( Y e. _V -> ( -. X _I Y <-> X =/= Y ) )
9	8	anbi2d 740	. . . 4 |- ( Y e. _V -> ( ( X A Y /\ -. X _I Y ) <-> ( X A Y /\ X =/= Y ) ) )
10	6, 9	syl5bb 272	. . 3 |- ( Y e. _V -> ( X ( A \ _I ) Y <-> ( X A Y /\ X =/= Y ) ) )
11	3, 5, 10	pm5.21nd 941	. 2 |- ( Rel A -> ( X ( A \ _I ) Y <-> ( X A Y /\ X =/= Y ) ) )
12		df-br 4654	. 2 |- ( X ( A \ _I ) Y <-> <. X , Y >. e. ( A \ _I ) )
13		df-br 4654	. . 3 |- ( X A Y <-> <. X , Y >. e. A )
14	13	anbi1i 731	. 2 |- ( ( X A Y /\ X =/= Y ) <-> ( <. X , Y >. e. A /\ X =/= Y ) )
15	11, 12, 14	3bitr3g 302	1 |- ( Rel A -> ( <. X , Y >. e. ( A \ _I ) <-> ( <. X , Y >. e. A /\ X =/= Y ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   <.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-ne 2795  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:  qtophaus  29903