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Theorem tfrlem15 6653
 Description: Lemma for transfinite recursion. Without assuming ax-rep 4320, 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 6647 . . 3 recs recs
32adantl 453 . 2 recs recs
41tfrlem13 6651 . . . 4 recs
5 simpr 448 . . . . 5 recs recs
6 resss 5170 . . . . . . . 8 recs recs
76a1i 11 . . . . . . 7 recs recs recs
81tfrlem6 6643 . . . . . . . . 9 recs
9 resdm 5184 . . . . . . . . 9 recs recs recs recs
108, 9ax-mp 8 . . . . . . . 8 recs recs recs
11 ssres2 5173 . . . . . . . 8 recs recs recs recs
1210, 11syl5eqssr 3393 . . . . . . 7 recs recs recs
137, 12eqssd 3365 . . . . . 6 recs recs recs
1413eleq1d 2502 . . . . 5 recs recs recs
155, 14syl5ibcom 212 . . . 4 recs recs recs
164, 15mtoi 171 . . 3 recs recs
171tfrlem8 6645 . . . 4 recs
18 eloni 4591 . . . . 5
1918adantr 452 . . . 4 recs
20 ordtri1 4614 . . . . 5 recs recs recs
2120con2bid 320 . . . 4 recs recs recs
2217, 19, 21sylancr 645 . . 3 recs recs recs
2316, 22mpbird 224 . 2 recs recs
243, 23impbida 806 1 recs recs
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cab 2422  wral 2705  wrex 2706  cvv 2956   wss 3320   word 4580  con0 4581   cdm 4878   cres 4880   wrel 4883   wfn 5449  cfv 5454  recscrecs 6632 This theorem is referenced by:  tfrlem16  6654  tfr2b  6657 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-recs 6633
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