Theorem frirr 4833

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 463	. . 3 |- ( ( R Fr A /\ B e. A ) -> R Fr A )
2		simpr 467	. . . 4 |- ( ( R Fr A /\ B e. A ) -> B e. A )
3	2	snssd 4130	. . 3 |- ( ( R Fr A /\ B e. A ) -> { B } C_ A )
4		snnzg 4102	. . . 4 |- ( B e. A -> { B } =/= (/) )
5	4	adantl 472	. . 3 |- ( ( R Fr A /\ B e. A ) -> { B } =/= (/) )
6		snex 4658	. . . 4 |- { B } e. _V
7	6	frc 4822	. . 3 |- ( ( R Fr A /\ { B } C_ A /\ { B } =/= (/) ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
8	1, 3, 5, 7	syl3anc 1276	. 2 |- ( ( R Fr A /\ B e. A ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
9		rabeq0 3766	. . . . . 6 |- ( { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R y )
10		breq2 4422	. . . . . . . 8 |- ( y = B -> ( x R y <-> x R B ) )
11	10	notbid 300	. . . . . . 7 |- ( y = B -> ( -. x R y <-> -. x R B ) )
12	11	ralbidv 2839	. . . . . 6 |- ( y = B -> ( A. x e. { B } -. x R y <-> A. x e. { B } -. x R B ) )
13	9, 12	syl5bb 265	. . . . 5 |- ( y = B -> ( { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R B ) )
14	13	rexsng 4019	. . . 4 |- ( B e. A -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R B ) )
15		breq1 4421	. . . . . 6 |- ( x = B -> ( x R B <-> B R B ) )
16	15	notbid 300	. . . . 5 |- ( x = B -> ( -. x R B <-> -. B R B ) )
17	16	ralsng 4018	. . . 4 |- ( B e. A -> ( A. x e. { B } -. x R B <-> -. B R B ) )
18	14, 17	bitrd 261	. . 3 |- ( B e. A -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> -. B R B ) )
19	18	adantl 472	. 2 |- ( ( R Fr A /\ B e. A ) -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> -. B R B ) )
20	8, 19	mpbid 215	1 |- ( ( R Fr A /\ B e. A ) -> -. B R B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   =/= wne 2633   A.wral 2749   E.wrex 2750   {crab 2753   C_ wss 3416   (/)c0 3743   {csn 3980   class class class wbr 4418   Fr wfr 4812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419  df-fr 4815

This theorem is referenced by:  efrirr  4837  predfrirr  5432  dfwe2  6640  bnj1417  29900  efrunt  30390  ifr0  36848