Theorem frirr 4861

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

Proof of Theorem frirr

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

Step	Hyp	Ref	Expression
1		simpl 457	. . 3
2		simpr 461	. . . 4
3	2	snssd 4175	. . 3
4		snnzg 4147	. . . 4
5	4	adantl 466	. . 3
6		snex 4693	. . . 4
7	6	frc 4850	. . 3
8	1, 3, 5, 7	syl3anc 1228	. 2
9		rabeq0 3807	. . . . . 6
10		breq2 4456	. . . . . . . 8
11	10	notbid 294	. . . . . . 7
12	11	ralbidv 2896	. . . . . 6
13	9, 12	syl5bb 257	. . . . 5
14	13	rexsng 4065	. . . 4
15		breq1 4455	. . . . . 6
16	15	notbid 294	. . . . 5
17	16	ralsng 4064	. . . 4
18	14, 17	bitrd 253	. . 3
19	18	adantl 466	. 2
20	8, 19	mpbid 210	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  {crab 2811  C_wss 3475  c0 3784  {csn 4029   class class class wbr 4452  Frwfr 4840

This theorem is referenced by:  efrirr  4865  dfwe2  6617  efrunt  29085  predfrirr  29278  ifr0  31359  bnj1417  34097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-fr 4843