Theorem frirr 5091

Description: A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
frirr	|- ( ( R Fr A /\ B e. A ) -> -. B R B )

Proof of Theorem frirr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( R Fr A /\ B e. A ) -> R Fr A )
2		snssi 4339	. . . 4 |- ( B e. A -> { B } C_ A )
3	2	adantl 482	. . 3 |- ( ( R Fr A /\ B e. A ) -> { B } C_ A )
4		snnzg 4308	. . . 4 |- ( B e. A -> { B } =/= (/) )
5	4	adantl 482	. . 3 |- ( ( R Fr A /\ B e. A ) -> { B } =/= (/) )
6		snex 4908	. . . 4 |- { B } e. _V
7	6	frc 5080	. . 3 |- ( ( R Fr A /\ { B } C_ A /\ { B } =/= (/) ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
8	1, 3, 5, 7	syl3anc 1326	. 2 |- ( ( R Fr A /\ B e. A ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
9		rabeq0 3957	. . . . . 6 |- ( { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R y )
10		breq2 4657	. . . . . . . 8 |- ( y = B -> ( x R y <-> x R B ) )
11	10	notbid 308	. . . . . . 7 |- ( y = B -> ( -. x R y <-> -. x R B ) )
12	11	ralbidv 2986	. . . . . 6 |- ( y = B -> ( A. x e. { B } -. x R y <-> A. x e. { B } -. x R B ) )
13	9, 12	syl5bb 272	. . . . 5 |- ( y = B -> ( { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R B ) )
14	13	rexsng 4219	. . . 4 |- ( B e. A -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R B ) )
15		breq1 4656	. . . . . 6 |- ( x = B -> ( x R B <-> B R B ) )
16	15	notbid 308	. . . . 5 |- ( x = B -> ( -. x R B <-> -. B R B ) )
17	16	ralsng 4218	. . . 4 |- ( B e. A -> ( A. x e. { B } -. x R B <-> -. B R B ) )
18	14, 17	bitrd 268	. . 3 |- ( B e. A -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> -. B R B ) )
19	18	adantl 482	. 2 |- ( ( R Fr A /\ B e. A ) -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> -. B R B ) )
20	8, 19	mpbid 222	1 |- ( ( R Fr A /\ B e. A ) -> -. B R B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   Fr wfr 5070

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-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-sbc 3436  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-fr 5073

This theorem is referenced by:  efrirr  5095  predfrirr  5709  dfwe2  6981  bnj1417  31109  efrunt  31590  ifr0  38654