Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem15 Structured version   Unicode version

Theorem tfrlem15 7079
 Description: Lemma for transfinite recursion. Without assuming ax-rep 4568, 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 7073 . . 3 recs recs
32adantl 466 . 2 recs recs
41tfrlem13 7077 . . . 4 recs
5 simpr 461 . . . . 5 recs recs
6 resss 5307 . . . . . . . 8 recs recs
76a1i 11 . . . . . . 7 recs recs recs
81tfrlem6 7069 . . . . . . . . 9 recs
9 resdm 5325 . . . . . . . . 9 recs recs recs recs
108, 9ax-mp 5 . . . . . . . 8 recs recs recs
11 ssres2 5310 . . . . . . . 8 recs recs recs recs
1210, 11syl5eqssr 3544 . . . . . . 7 recs recs recs
137, 12eqssd 3516 . . . . . 6 recs recs recs
1413eleq1d 2526 . . . . 5 recs recs recs
155, 14syl5ibcom 220 . . . 4 recs recs recs
164, 15mtoi 178 . . 3 recs recs
171tfrlem8 7071 . . . 4 recs
18 eloni 4897 . . . . 5
1918adantr 465 . . . 4 recs
20 ordtri1 4920 . . . . 5 recs recs recs
2120con2bid 329 . . . 4 recs recs recs
2217, 19, 21sylancr 663 . . 3 recs recs recs
2316, 22mpbird 232 . 2 recs recs
243, 23impbida 832 1 recs recs
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wa 369   wceq 1395   wcel 1819  cab 2442  wral 2807  wrex 2808  cvv 3109   wss 3471   word 4886  con0 4887   cdm 5008   cres 5010   wrel 5013   wfn 5589  cfv 5594  recscrecs 7059 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-recs 7060 This theorem is referenced by:  tfrlem16  7080  tfr2b  7083
 Copyright terms: Public domain W3C validator