Theorem frirr 4830

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 458	. . 3 |- ( ( R Fr A /\ B e. A ) -> R Fr A )
2		simpr 462	. . . 4 |- ( ( R Fr A /\ B e. A ) -> B e. A )
3	2	snssd 4145	. . 3 |- ( ( R Fr A /\ B e. A ) -> { B } C_ A )
4		snnzg 4117	. . . 4 |- ( B e. A -> { B } =/= (/) )
5	4	adantl 467	. . 3 |- ( ( R Fr A /\ B e. A ) -> { B } =/= (/) )
6		snex 4662	. . . 4 |- { B } e. _V
7	6	frc 4819	. . 3 |- ( ( R Fr A /\ { B } C_ A /\ { B } =/= (/) ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
8	1, 3, 5, 7	syl3anc 1264	. 2 |- ( ( R Fr A /\ B e. A ) -> E. y e. { B } { x e. { B } | x R y } = (/) )
9		rabeq0 3784	. . . . . 6 |- ( { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R y )
10		breq2 4427	. . . . . . . 8 |- ( y = B -> ( x R y <-> x R B ) )
11	10	notbid 295	. . . . . . 7 |- ( y = B -> ( -. x R y <-> -. x R B ) )
12	11	ralbidv 2861	. . . . . 6 |- ( y = B -> ( A. x e. { B } -. x R y <-> A. x e. { B } -. x R B ) )
13	9, 12	syl5bb 260	. . . . 5 |- ( y = B -> ( { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R B ) )
14	13	rexsng 4035	. . . 4 |- ( B e. A -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> A. x e. { B } -. x R B ) )
15		breq1 4426	. . . . . 6 |- ( x = B -> ( x R B <-> B R B ) )
16	15	notbid 295	. . . . 5 |- ( x = B -> ( -. x R B <-> -. B R B ) )
17	16	ralsng 4034	. . . 4 |- ( B e. A -> ( A. x e. { B } -. x R B <-> -. B R B ) )
18	14, 17	bitrd 256	. . 3 |- ( B e. A -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> -. B R B ) )
19	18	adantl 467	. 2 |- ( ( R Fr A /\ B e. A ) -> ( E. y e. { B } { x e. { B } | x R y } = (/) <-> -. B R B ) )
20	8, 19	mpbid 213	1 |- ( ( R Fr A /\ B e. A ) -> -. B R B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1872   =/= wne 2614   A.wral 2771   E.wrex 2772   {crab 2775   C_ wss 3436   (/)c0 3761   {csn 3998   class class class wbr 4423   Fr wfr 4809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-fr 4812

This theorem is referenced by:  efrirr  4834  predfrirr  5428  dfwe2  6622  bnj1417  29858  efrunt  30348  ifr0  36773