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Theorem tfrlem10 6289
 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 6293, 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 5391 . . . . . . 7 recs
2 funsng 5155 . . . . . . 7 recs recs recs recs
31, 2mpan2 655 . . . . . 6 recs recs recs
4 tfrlem.1 . . . . . . 7
54tfrlem7 6285 . . . . . 6 recs
63, 5jctil 525 . . . . 5 recs recs recs recs
71dmsnop 5053 . . . . . . 7 recs recs recs
87ineq2i 3275 . . . . . 6 recs recs recs recs recs
94tfrlem8 6286 . . . . . . 7 recs
10 orddisj 4323 . . . . . . 7 recs recs recs
119, 10ax-mp 10 . . . . . 6 recs recs
128, 11eqtri 2273 . . . . 5 recs recs recs
13 funun 5153 . . . . 5 recs recs recs recs recs recs recs recs recs
146, 12, 13sylancl 646 . . . 4 recs recs recs recs
157uneq2i 3236 . . . . 5 recs recs recs recs recs
16 dmun 4792 . . . . 5 recs recs recs recs recs recs
17 df-suc 4291 . . . . 5 recs recs recs
1815, 16, 173eqtr4i 2283 . . . 4 recs recs recs recs
1914, 18jctir 526 . . 3 recs recs recs recs recs recs recs recs
20 df-fn 4603 . . 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 5195 . 2 recs recs recs recs recs
2421, 23sylibr 205 1 recs recs
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360   wceq 1619   wcel 1621  cab 2239  wral 2509  wrex 2510  cvv 2727   cun 3076   cin 3077  c0 3362  csn 3544  cop 3547   word 4284  con0 4285   csuc 4287   cdm 4580   cres 4582   wfun 4586   wfn 4587  cfv 4592  recscrecs 6273 This theorem is referenced by:  tfrlem11  6290  tfrlem12  6291  tfrlem13  6292 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-fv 4608  df-recs 6274
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