Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem13 Unicode version

Theorem tfrlem13 6406
 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 6400 . . 3 recs
3 ordirr 4410 . . 3 recs recs recs
42, 3ax-mp 8 . 2 recs recs
5 eqid 2283 . . . . 5 recs recs recs recs recs recs
61, 5tfrlem12 6405 . . . 4 recs recs recs recs
7 elssuni 3855 . . . . 5 recs recs recs recs recs recs
81recsfval 6397 . . . . 5 recs
97, 8syl6sseqr 3225 . . . 4 recs recs recs recs recs recs recs
10 dmss 4878 . . . 4 recs recs recs recs recs recs recs recs
116, 9, 103syl 18 . . 3 recs recs recs recs recs
122a1i 10 . . . . . 6 recs recs
13 dmexg 4939 . . . . . 6 recs recs
14 elon2 4403 . . . . . 6 recs recs recs
1512, 13, 14sylanbrc 645 . . . . 5 recs recs
16 sucidg 4470 . . . . 5 recs recs recs
1715, 16syl 15 . . . 4 recs recs recs
181, 5tfrlem10 6403 . . . . 5 recs recs recs recs recs
19 fndm 5343 . . . . 5 recs recs recs recs recs recs recs recs
2015, 18, 193syl 18 . . . 4 recs recs recs recs recs
2117, 20eleqtrrd 2360 . . 3 recs recs recs recs recs
2211, 21sseldd 3181 . 2 recs recs recs
234, 22mto 167 1 recs
 Colors of variables: wff set class Syntax hints:   wn 3   wa 358   wceq 1623   wcel 1684  cab 2269  wral 2543  wrex 2544  cvv 2788   cun 3150   wss 3152  csn 3640  cop 3643  cuni 3827   word 4391  con0 4392   csuc 4394   cdm 4689   cres 4691   wfn 5250  cfv 5255  recscrecs 6387 This theorem is referenced by:  tfrlem14  6407  tfrlem15  6408  tfrlem16  6409  tfr2b  6412 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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  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-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-recs 6388
 Copyright terms: Public domain W3C validator