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Theorem hsmex 8317
 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 7563. (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 4219 . . . . 5
21ralbidv 2727 . . . 4
32rabbidv 2950 . . 3
43eleq1d 2504 . 2
5 vex 2961 . . 3
6 eqid 2438 . . 3 har har har har
7 rdgeq2 6673 . . . . . 6
8 unieq 4026 . . . . . . . 8
98cbvmptv 4303 . . . . . . 7
10 rdgeq1 6672 . . . . . . 7
119, 10ax-mp 5 . . . . . 6
127, 11syl6eq 2486 . . . . 5
1312reseq1d 5148 . . . 4
1413cbvmptv 4303 . . 3
15 eqid 2438 . . 3
16 eqid 2438 . . 3 OrdIso OrdIso
175, 6, 14, 15, 16hsmexlem6 8316 . 2
184, 17vtoclg 3013 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  wral 2707  crab 2711  cvv 2958  cpw 3801  csn 3816  cuni 4017   class class class wbr 4215   cmpt 4269   cep 4495  con0 4584  com 4848   cxp 4879   cres 4883  cima 4884  cfv 5457  crdg 6670   cdom 7110  OrdIsocoi 7481  harchar 7527  ctc 7678  cr1 7691  crnk 7692 This theorem is referenced by:  hsmex2  8318 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-1st 6352  df-2nd 6353  df-riota 6552  df-smo 6611  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-oi 7482  df-har 7529  df-wdom 7530  df-tc 7679  df-r1 7693  df-rank 7694
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