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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.
StepHypRef Expression
1 simpl 457 . . 3
2 simpr 461 . . . 4
32snssd 4175 . . 3
4 snnzg 4147 . . . 4
54adantl 466 . . 3
6 snex 4693 . . . 4
76frc 4850 . . 3
81, 3, 5, 7syl3anc 1228 . 2
9 rabeq0 3807 . . . . . 6
10 breq2 4456 . . . . . . . 8
1110notbid 294 . . . . . . 7
1211ralbidv 2896 . . . . . 6
139, 12syl5bb 257 . . . . 5
1413rexsng 4065 . . . 4
15 breq1 4455 . . . . . 6
1615notbid 294 . . . . 5
1716ralsng 4064 . . . 4
1814, 17bitrd 253 . . 3
1918adantl 466 . 2
208, 19mpbid 210 1
Colors of variables: wff setvar class
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
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