Theorem frirr 4770

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 455	. . 3 |- ( ( R Fr A /\ B e. A ) -> R Fr A )
2		simpr 459	. . . 4 |- ( ( R Fr A /\ B e. A ) -> B e. A )
3	2	snssd 4089	. . 3 |- ( ( R Fr A /\ B e. A ) -> { B } C_ A )
4		snnzg 4061	. . . 4 |- ( B e. A -> { B } =/= (/) )
5	4	adantl 464	. . 3 |- ( ( R Fr A /\ B e. A ) -> { B } =/= (/) )
6		snex 4603	. . . 4 |- { B } e. _V
7	6	frc 4759	. . 3 |- ( ( R Fr A /\ { B } C_ A /\ { B } =/= (/) ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
8	1, 3, 5, 7	syl3anc 1226	. 2 |- ( ( R Fr A /\ B e. A ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
9		rabeq0 3734	. . . . . 6 |- ( { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R y )
10		breq2 4371	. . . . . . . 8 |- ( y = B -> ( x R y <-> x R B ) )
11	10	notbid 292	. . . . . . 7 |- ( y = B -> ( -. x R y <-> -. x R B ) )
12	11	ralbidv 2821	. . . . . 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 3980	. . . 4 |- ( B e. A -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R B ) )
15		breq1 4370	. . . . . 6 |- ( x = B -> ( x R B <-> B R B ) )
16	15	notbid 292	. . . . 5 |- ( x = B -> ( -. x R B <-> -. B R B ) )
17	16	ralsng 3979	. . . 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 464	. 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 367   = wceq 1399   e. wcel 1826   =/= wne 2577   A.wral 2732   E.wrex 2733   {crab 2736   C_ wss 3389   (/)c0 3711   {csn 3944   class class class wbr 4367   Fr wfr 4749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-fr 4752

This theorem is referenced by:  efrirr  4774  dfwe2  6516  efrunt  29251  predfrirr  29443  ifr0  31527  bnj1417  34444