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Theorem tfrlem16 6295
 Description: Lemma for finite recursion. Without assuming ax-rep 4028, 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
StepHypRef Expression
1 tfrlem.1 . . . 4
21tfrlem8 6286 . . 3 recs
3 ordzsl 4527 . . 3 recs recs recs recs
42, 3mpbi 201 . 2 recs recs recs
5 res0 4866 . . . . . . 7 recs
6 0ex 4047 . . . . . . 7
75, 6eqeltri 2323 . . . . . 6 recs
8 0elon 4338 . . . . . . 7
91tfrlem15 6294 . . . . . . 7 recs recs
108, 9ax-mp 10 . . . . . 6 recs recs
117, 10mpbir 202 . . . . 5 recs
12 n0i 3367 . . . . 5 recs recs
1311, 12ax-mp 10 . . . 4 recs
1413pm2.21i 125 . . 3 recs recs
151tfrlem13 6292 . . . . 5 recs
16 simpr 449 . . . . . . . . . 10 recs recs
17 df-suc 4291 . . . . . . . . . 10
1816, 17syl6eq 2301 . . . . . . . . 9 recs recs
1918reseq2d 4862 . . . . . . . 8 recs recs recs recs
201tfrlem6 6284 . . . . . . . . 9 recs
21 resdm 4900 . . . . . . . . 9 recs recs recs recs
2220, 21ax-mp 10 . . . . . . . 8 recs recs recs
23 resundi 4876 . . . . . . . 8 recs recs recs
2419, 22, 233eqtr3g 2308 . . . . . . 7 recs recs recs recs
25 vex 2730 . . . . . . . . . . 11
2625sucid 4364 . . . . . . . . . 10
2726, 16syl5eleqr 2340 . . . . . . . . 9 recs recs
281tfrlem9a 6288 . . . . . . . . 9 recs recs
2927, 28syl 17 . . . . . . . 8 recs recs
30 snex 4110 . . . . . . . . 9 recs
311tfrlem7 6285 . . . . . . . . . 10 recs
32 funressn 5558 . . . . . . . . . 10 recs recs recs
3331, 32ax-mp 10 . . . . . . . . 9 recs recs
3430, 33ssexi 4056 . . . . . . . 8 recs
35 unexg 4412 . . . . . . . 8 recs recs recs recs
3629, 34, 35sylancl 646 . . . . . . 7 recs recs recs
3724, 36eqeltrd 2327 . . . . . 6 recs recs
3837rexlimiva 2624 . . . . 5 recs recs
3915, 38mto 169 . . . 4 recs
4039pm2.21i 125 . . 3 recs recs
41 id 21 . . 3 recs recs
4214, 40, 413jaoi 1250 . 2 recs recs recs recs
434, 42ax-mp 10 1 recs
 Colors of variables: wff set class Syntax hints:   wn 5   wb 178   wa 360   w3o 938   wceq 1619   wcel 1621  cab 2239  wral 2509  wrex 2510  cvv 2727   cun 3076   wss 3078  c0 3362  csn 3544  cop 3547   word 4284  con0 4285   wlim 4286   csuc 4287   cdm 4580   cres 4582   wrel 4585   wfun 4586   wfn 4587  cfv 4592  recscrecs 6273 This theorem is referenced by:  tfr1a  6296 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-reu 2515  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-pw 3532  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-lim 4290  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-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-recs 6274
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