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Theorem tfrlem15 6294
 Description: Lemma for transfinite recursion. Without assuming ax-rep 4028, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 14-Nov-2014.)
Hypothesis
Ref Expression
tfrlem.1
Assertion
Ref Expression
tfrlem15 recs recs
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem15
StepHypRef Expression
1 tfrlem.1 . . . 4
21tfrlem9a 6288 . . 3 recs recs
32adantl 454 . 2 recs recs
41tfrlem13 6292 . . . 4 recs
5 simpr 449 . . . . 5 recs recs
6 resss 4886 . . . . . . . 8 recs recs
76a1i 12 . . . . . . 7 recs recs recs
81tfrlem6 6284 . . . . . . . . 9 recs
9 resdm 4900 . . . . . . . . 9 recs recs recs recs
108, 9ax-mp 10 . . . . . . . 8 recs recs recs
11 ssres2 4889 . . . . . . . 8 recs recs recs recs
1210, 11syl5eqssr 3144 . . . . . . 7 recs recs recs
137, 12eqssd 3117 . . . . . 6 recs recs recs
1413eleq1d 2319 . . . . 5 recs recs recs
155, 14syl5ibcom 213 . . . 4 recs recs recs
164, 15mtoi 171 . . 3 recs recs
171tfrlem8 6286 . . . 4 recs
18 eloni 4295 . . . . 5
1918adantr 453 . . . 4 recs
20 ordtri1 4318 . . . . 5 recs recs recs
2120con2bid 321 . . . 4 recs recs recs
2217, 19, 21sylancr 647 . . 3 recs recs recs
2316, 22mpbird 225 . 2 recs recs
243, 23impbida 808 1 recs recs
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wb 178   wa 360   wceq 1619   wcel 1621  cab 2239  wral 2509  wrex 2510  cvv 2727   wss 3078   word 4284  con0 4285   cdm 4580   cres 4582   wrel 4585   wfn 4587  cfv 4592  recscrecs 6273 This theorem is referenced by:  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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