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Theorem tfrlem16 6409
 Description: Lemma for finite recursion. Without assuming ax-rep 4131, we can show that the domain of the constructed function is a limit ordinal, and hence contains all the finite ordinals. (Contributed by Mario Carneiro, 14-Nov-2014.)
Hypothesis
Ref Expression
tfrlem.1
Assertion
Ref Expression
tfrlem16 recs
Distinct variable group:   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem16
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . 4
21tfrlem8 6400 . . 3 recs
3 ordzsl 4636 . . 3 recs recs recs recs
42, 3mpbi 199 . 2 recs recs recs
5 res0 4959 . . . . . . 7 recs
6 0ex 4150 . . . . . . 7
75, 6eqeltri 2353 . . . . . 6 recs
8 0elon 4445 . . . . . . 7
91tfrlem15 6408 . . . . . . 7 recs recs
108, 9ax-mp 8 . . . . . 6 recs recs
117, 10mpbir 200 . . . . 5 recs
12 n0i 3460 . . . . 5 recs recs
1311, 12ax-mp 8 . . . 4 recs
1413pm2.21i 123 . . 3 recs recs
151tfrlem13 6406 . . . . 5 recs
16 simpr 447 . . . . . . . . . 10 recs recs
17 df-suc 4398 . . . . . . . . . 10
1816, 17syl6eq 2331 . . . . . . . . 9 recs recs
1918reseq2d 4955 . . . . . . . 8 recs recs recs recs
201tfrlem6 6398 . . . . . . . . 9 recs
21 resdm 4993 . . . . . . . . 9 recs recs recs recs
2220, 21ax-mp 8 . . . . . . . 8 recs recs recs
23 resundi 4969 . . . . . . . 8 recs recs recs
2419, 22, 233eqtr3g 2338 . . . . . . 7 recs recs recs recs
25 vex 2791 . . . . . . . . . . 11
2625sucid 4471 . . . . . . . . . 10
2726, 16syl5eleqr 2370 . . . . . . . . 9 recs recs
281tfrlem9a 6402 . . . . . . . . 9 recs recs
2927, 28syl 15 . . . . . . . 8 recs recs
30 snex 4216 . . . . . . . . 9 recs
311tfrlem7 6399 . . . . . . . . . 10 recs
32 funressn 5706 . . . . . . . . . 10 recs recs recs
3331, 32ax-mp 8 . . . . . . . . 9 recs recs
3430, 33ssexi 4159 . . . . . . . 8 recs
35 unexg 4521 . . . . . . . 8 recs recs recs recs
3629, 34, 35sylancl 643 . . . . . . 7 recs recs recs
3724, 36eqeltrd 2357 . . . . . 6 recs recs
3837rexlimiva 2662 . . . . 5 recs recs
3915, 38mto 167 . . . 4 recs
4039pm2.21i 123 . . 3 recs recs
41 id 19 . . 3 recs recs
4214, 40, 413jaoi 1245 . 2 recs recs recs recs
434, 42ax-mp 8 1 recs
 Colors of variables: wff set class Syntax hints:   wn 3   wb 176   wa 358   w3o 933   wceq 1623   wcel 1684  cab 2269  wral 2543  wrex 2544  cvv 2788   cun 3150   wss 3152  c0 3455  csn 3640  cop 3643   word 4391  con0 4392   wlim 4393   csuc 4394   cdm 4689   cres 4691   wrel 4694   wfun 5249   wfn 5250  cfv 5255  recscrecs 6387 This theorem is referenced by:  tfr1a  6410 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-reu 2550  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-pw 3627  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-lim 4397  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-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388
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