Theorem frirr 4849

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 457	. . 3 |- ( ( R Fr A /\ B e. A ) -> R Fr A )
2		simpr 461	. . . 4 |- ( ( R Fr A /\ B e. A ) -> B e. A )
3	2	snssd 4165	. . 3 |- ( ( R Fr A /\ B e. A ) -> { B } C_ A )
4		snnzg 4137	. . . 4 |- ( B e. A -> { B } =/= (/) )
5	4	adantl 466	. . 3 |- ( ( R Fr A /\ B e. A ) -> { B } =/= (/) )
6		snex 4681	. . . 4 |- { B } e. _V
7	6	frc 4838	. . 3 |- ( ( R Fr A /\ { B } C_ A /\ { B } =/= (/) ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
8	1, 3, 5, 7	syl3anc 1223	. 2 |- ( ( R Fr A /\ B e. A ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
9		rabeq0 3800	. . . . . 6 |- ( { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R y )
10		breq2 4444	. . . . . . . 8 |- ( y = B -> ( x R y <-> x R B ) )
11	10	notbid 294	. . . . . . 7 |- ( y = B -> ( -. x R y <-> -. x R B ) )
12	11	ralbidv 2896	. . . . . 6 |- ( y = B -> ( A. x e. { B } -. x R y <-> A. x e. { B } -. x R B ) )
13	9, 12	syl5bb 257	. . . . 5 |- ( y = B -> ( { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R B ) )
14	13	rexsng 4056	. . . 4 |- ( B e. A -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R B ) )
15		breq1 4443	. . . . . 6 |- ( x = B -> ( x R B <-> B R B ) )
16	15	notbid 294	. . . . 5 |- ( x = B -> ( -. x R B <-> -. B R B ) )
17	16	ralsng 4055	. . . 4 |- ( B e. A -> ( A. x e. { B } -. x R B <-> -. B R B ) )
18	14, 17	bitrd 253	. . 3 |- ( B e. A -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> -. B R B ) )
19	18	adantl 466	. 2 |- ( ( R Fr A /\ B e. A ) -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> -. B R B ) )
20	8, 19	mpbid 210	1 |- ( ( R Fr A /\ B e. A ) -> -. B R B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   e. wcel 1762   =/= wne 2655   A.wral 2807   E.wrex 2808   {crab 2811   C_ wss 3469   (/)c0 3778   {csn 4020   class class class wbr 4440   Fr wfr 4828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-fr 4831

This theorem is referenced by:  efrirr  4853  dfwe2  6588  efrunt  28546  predfrirr  28841  ifr0  30892  bnj1417  33051