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Theorem fr2nr 4371
 Description: A well-founded relation has no 2-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
fr2nr

Proof of Theorem fr2nr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prex 4217 . . . . . . 7
21a1i 10 . . . . . 6
3 simpl 443 . . . . . 6
4 prssi 3771 . . . . . . 7
54adantl 452 . . . . . 6
6 prnzg 3746 . . . . . . 7
76ad2antrl 708 . . . . . 6
8 fri 4355 . . . . . 6
92, 3, 5, 7, 8syl22anc 1183 . . . . 5
10 breq2 4027 . . . . . . . . 9
1110notbid 285 . . . . . . . 8
1211ralbidv 2563 . . . . . . 7
13 breq2 4027 . . . . . . . . 9
1413notbid 285 . . . . . . . 8
1514ralbidv 2563 . . . . . . 7
1612, 15rexprg 3683 . . . . . 6
1716adantl 452 . . . . 5
189, 17mpbid 201 . . . 4
19 prid2g 3733 . . . . . . 7
2019ad2antll 709 . . . . . 6
21 breq1 4026 . . . . . . . 8
2221notbid 285 . . . . . . 7
2322rspcv 2880 . . . . . 6
2420, 23syl 15 . . . . 5
25 prid1g 3732 . . . . . . 7
2625ad2antrl 708 . . . . . 6
27 breq1 4026 . . . . . . . 8
2827notbid 285 . . . . . . 7
2928rspcv 2880 . . . . . 6
3026, 29syl 15 . . . . 5
3124, 30orim12d 811 . . . 4
3218, 31mpd 14 . . 3
3332orcomd 377 . 2
34 ianor 474 . 2
3533, 34sylibr 203 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wo 357   wa 358   wceq 1623   wcel 1684   wne 2446  wral 2543  wrex 2544  cvv 2788   wss 3152  c0 3455  cpr 3641   class class class wbr 4023   wfr 4349 This theorem is referenced by:  efrn2lp  4375  dfwe2  4573 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-fr 4352
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