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Theorem hsmex 8844
 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 8052. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Assertion
Ref Expression
hsmex
Distinct variable group:   ,,
Allowed substitution hints:   (,)

Proof of Theorem hsmex
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4399 . . . . 5
21ralbidv 2843 . . . 4
32rabbidv 3051 . . 3
43eleq1d 2471 . 2
5 vex 3062 . . 3
6 eqid 2402 . . 3 har har har har
7 rdgeq2 7115 . . . . . 6
8 unieq 4199 . . . . . . . 8
98cbvmptv 4487 . . . . . . 7
10 rdgeq1 7114 . . . . . . 7
119, 10ax-mp 5 . . . . . 6
127, 11syl6eq 2459 . . . . 5
1312reseq1d 5093 . . . 4
1413cbvmptv 4487 . . 3
15 eqid 2402 . . 3
16 eqid 2402 . . 3 OrdIso OrdIso
175, 6, 14, 15, 16hsmexlem6 8843 . 2
184, 17vtoclg 3117 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1405   wcel 1842  wral 2754  crab 2758  cvv 3059  cpw 3955  csn 3972  cuni 4191   class class class wbr 4395   cmpt 4453   cep 4732   cxp 4821   cres 4825  cima 4826  con0 5410  cfv 5569  com 6683  crdg 7112   cdom 7552  OrdIsocoi 7968  harchar 8016  ctc 8199  cr1 8212  crnk 8213 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-smo 7050  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-oi 7969  df-har 8018  df-wdom 8019  df-tc 8200  df-r1 8214  df-rank 8215 This theorem is referenced by:  hsmex2  8845
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