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Theorem hsmex 8833
Description: The collection of hereditarily size-limited well-founded sets comprise a set. The proof is that of Randall Holmes at http://math.boisestate.edu/~holmes/holmes/hereditary.pdf, with modifications to use Hartogs' theorem instead of the weak variant (inconsequentially weakening some intermediate results), and making the well-foundedness condition explicit to avoid a direct dependence on ax-reg 8039. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Assertion
Ref Expression
hsmex
Distinct variable group:   , ,

Proof of Theorem hsmex
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4456 . . . . 5
21ralbidv 2896 . . . 4
32rabbidv 3101 . . 3
43eleq1d 2526 . 2
5 vex 3112 . . 3
6 eqid 2457 . . 3
7 rdgeq2 7097 . . . . . 6
8 unieq 4257 . . . . . . . 8
98cbvmptv 4543 . . . . . . 7
10 rdgeq1 7096 . . . . . . 7
119, 10ax-mp 5 . . . . . 6
127, 11syl6eq 2514 . . . . 5
1312reseq1d 5277 . . . 4
1413cbvmptv 4543 . . 3
15 eqid 2457 . . 3
16 eqid 2457 . . 3
175, 6, 14, 15, 16hsmexlem6 8832 . 2
184, 17vtoclg 3167 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  A.wral 2807  {crab 2811   cvv 3109  ~Pcpw 4012  {csn 4029  U.cuni 4249   class class class wbr 4452  e.cmpt 4510   cep 4794   con0 4883  X.cxp 5002  |`cres 5006  "cima 5007  `cfv 5593   com 6700  reccrdg 7094   cdom 7534  OrdIsocoi 7955   char 8003   ctc 8188   cr1 8201   crnk 8202
This theorem is referenced by:  hsmex2  8834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-inf2 8079
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 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-int 4287  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6257  df-om 6701  df-1st 6800  df-2nd 6801  df-smo 7036  df-recs 7061  df-rdg 7095  df-er 7330  df-en 7537  df-dom 7538  df-sdom 7539  df-oi 7956  df-har 8005  df-wdom 8006  df-tc 8189  df-r1 8203  df-rank 8204
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