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Theorem tfrlem13 6292
 Description: Lemma for transfinite recursion. If recs is a set function, then is acceptable, and thus a subset of recs, but is bigger than recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypothesis
Ref Expression
tfrlem.1
Assertion
Ref Expression
tfrlem13 recs
Distinct variable group:   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem13
StepHypRef Expression
1 tfrlem.1 . . . 4
21tfrlem8 6286 . . 3 recs
3 ordirr 4303 . . 3 recs recs recs
42, 3ax-mp 10 . 2 recs recs
5 eqid 2253 . . . . 5 recs recs recs recs recs recs
61, 5tfrlem12 6291 . . . 4 recs recs recs recs
7 elssuni 3753 . . . . 5 recs recs recs recs recs recs
81recsfval 6283 . . . . 5 recs
97, 8syl6sseqr 3146 . . . 4 recs recs recs recs recs recs recs
10 dmss 4785 . . . 4 recs recs recs recs recs recs recs recs
116, 9, 103syl 20 . . 3 recs recs recs recs recs
122a1i 12 . . . . . 6 recs recs
13 dmexg 4846 . . . . . 6 recs recs
14 elon2 4296 . . . . . 6 recs recs recs
1512, 13, 14sylanbrc 648 . . . . 5 recs recs
16 sucidg 4363 . . . . 5 recs recs recs
1715, 16syl 17 . . . 4 recs recs recs
181, 5tfrlem10 6289 . . . . 5 recs recs recs recs recs
19 fndm 5200 . . . . 5 recs recs recs recs recs recs recs recs
2015, 18, 193syl 20 . . . 4 recs recs recs recs recs
2117, 20eleqtrrd 2330 . . 3 recs recs recs recs recs
2211, 21sseldd 3104 . 2 recs recs recs
234, 22mto 169 1 recs
 Colors of variables: wff set class Syntax hints:   wn 5   wa 360   wceq 1619   wcel 1621  cab 2239  wral 2509  wrex 2510  cvv 2727   cun 3076   wss 3078  csn 3544  cop 3547  cuni 3727   word 4284  con0 4285   csuc 4287   cdm 4580   cres 4582   wfn 4587  cfv 4592  recscrecs 6273 This theorem is referenced by:  tfrlem14  6293  tfrlem15  6294  tfrlem16  6295  tfr2b  6298 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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