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Theorem tfrlem10 7075
Description: Lemma for transfinite recursion. We define class by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to, . Using this assumption we will prove facts about that will lead to a contradiction in tfrlem14 7079, thus showing the domain of recs does in fact equal . Here we show (under the false assumption) that is a function extending the domain of by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
tfrlem.1
tfrlem.3
Assertion
Ref Expression
tfrlem10
Distinct variable groups:   , , ,   , , ,

Proof of Theorem tfrlem10
StepHypRef Expression
1 fvex 5881 . . . . . . 7
2 funsng 5639 . . . . . . 7
31, 2mpan2 671 . . . . . 6
4 tfrlem.1 . . . . . . 7
54tfrlem7 7071 . . . . . 6
63, 5jctil 537 . . . . 5
71dmsnop 5487 . . . . . . 7
87ineq2i 3696 . . . . . 6
94tfrlem8 7072 . . . . . . 7
10 orddisj 4921 . . . . . . 7
119, 10ax-mp 5 . . . . . 6
128, 11eqtri 2486 . . . . 5
13 funun 5635 . . . . 5
146, 12, 13sylancl 662 . . . 4
157uneq2i 3654 . . . . 5
16 dmun 5214 . . . . 5
17 df-suc 4889 . . . . 5
1815, 16, 173eqtr4i 2496 . . . 4
1914, 18jctir 538 . . 3
20 df-fn 5596 . . 3
2119, 20sylibr 212 . 2
22 tfrlem.3 . . 3
2322fneq1i 5680 . 2
2421, 23sylibr 212 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  E.wrex 2808   cvv 3109  u.cun 3473  i^icin 3474   c0 3784  {csn 4029  <.cop 4035  Ordword 4882   con0 4883  succsuc 4885  domcdm 5004  |`cres 5006  Funwfun 5587  Fnwfn 5588  `cfv 5593  recscrecs 7060
This theorem is referenced by:  tfrlem11  7076  tfrlem12  7077  tfrlem13  7078
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-8 1820  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-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  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-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601  df-recs 7061
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