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Theorem tfrlem10 6677
 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 6681, 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 recs by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
tfrlem.1
tfrlem.3 recs recs recs
Assertion
Ref Expression
tfrlem10 recs recs
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem10
StepHypRef Expression
1 fvex 5771 . . . . . . 7 recs
2 funsng 5526 . . . . . . 7 recs recs recs recs
31, 2mpan2 654 . . . . . 6 recs recs recs
4 tfrlem.1 . . . . . . 7
54tfrlem7 6673 . . . . . 6 recs
63, 5jctil 525 . . . . 5 recs recs recs recs
71dmsnop 5373 . . . . . . 7 recs recs recs
87ineq2i 3525 . . . . . 6 recs recs recs recs recs
94tfrlem8 6674 . . . . . . 7 recs
10 orddisj 4648 . . . . . . 7 recs recs recs
119, 10ax-mp 5 . . . . . 6 recs recs
128, 11eqtri 2462 . . . . 5 recs recs recs
13 funun 5524 . . . . 5 recs recs recs recs recs recs recs recs recs
146, 12, 13sylancl 645 . . . 4 recs recs recs recs
157uneq2i 3484 . . . . 5 recs recs recs recs recs
16 dmun 5105 . . . . 5 recs recs recs recs recs recs
17 df-suc 4616 . . . . 5 recs recs recs
1815, 16, 173eqtr4i 2472 . . . 4 recs recs recs recs
1914, 18jctir 526 . . 3 recs recs recs recs recs recs recs recs
20 df-fn 5486 . . 3 recs recs recs recs recs recs recs recs recs recs recs
2119, 20sylibr 205 . 2 recs recs recs recs recs
22 tfrlem.3 . . 3 recs recs recs
2322fneq1i 5568 . 2 recs recs recs recs recs
2421, 23sylibr 205 1 recs recs
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1727  cab 2428  wral 2711  wrex 2712  cvv 2962   cun 3304   cin 3305  c0 3613  csn 3838  cop 3841   word 4609  con0 4610   csuc 4612   cdm 4907   cres 4909   wfun 5477   wfn 5478  cfv 5483  recscrecs 6661 This theorem is referenced by:  tfrlem11  6678  tfrlem12  6679  tfrlem13  6680 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-suc 4616  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-fv 5491  df-recs 6662
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