Theorem hta 8336

Description: A ZFC emulation of Hilbert's transfinite axiom. The set has the properties of Hilbert's epsilon, except that it also depends on a well-ordering . This theorem arose from discussions with Raph Levien on 5-Mar-2004 about translating the HOL proof language, which uses Hilbert's epsilon. See http://us.metamath.org/downloads/choice.txt (copy of obsolete link http://ghilbert.org/choice.txt) and http://us.metamath.org/downloads/megillaward2005he.pdf.

Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem differs from Hilbert's transfinite axiom described on that page in that it requires as an antecedent. Class collects the sets of the least rank for which (x) is true. Class , which emulates the epsilon, is the minimum element in a well-ordering on .

If a well-ordering on can be expressed in a closed form, as might be the case if we are working with say natural numbers, we can eliminate the antecedent with modus ponens, giving us the exact equivalent of Hilbert's transfinite axiom. Otherwise, we replace with a dummy setvar variable, say , and attach as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point, (which will have as a free variable) will no longer be present, and we can eliminate by applying exlimiv 1722 and weth 8896, using scottexs 8326 to establish the existence of .

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 8335. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)

Hypotheses

Ref	Expression
hta.1
hta.2

Assertion

Ref	Expression
hta

Distinct variable groups:   ,   ,,   ,   ,,

Proof of Theorem hta

Step	Hyp	Ref	Expression
1		19.8a 1857	. . 3
2		scott0s 8327	. . . 4
3		hta.1	. . . . 5
4	3	neeq1i 2742	. . . 4
5	2, 4	bitr4i 252	. . 3
6	1, 5	sylib 196	. 2
7		scottexs 8326	. . . . 5
8	3, 7	eqeltri 2541	. . . 4
9		hta.2	. . . 4
10	8, 9	htalem 8335	. . 3
11	10	ex 434	. 2
12		simpl 457	. . . . . 6
13	12	ss2abi 3571	. . . . 5
14	3, 13	eqsstri 3533	. . . 4
15	14	sseli 3499	. . 3
16		df-sbc 3328	. . 3
17	15, 16	sylibr 212	. 2
18	6, 11, 17	syl56 34	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  =/=wne 2652  A.wral 2807  cvv 3109  [.wsbc 3327  C_wss 3475  c0 3784   class class class wbr 4452  Wewwe 4842  `cfv 5593  iota_crio 6256  crnk 8202

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-reg 8039  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-iin 4333  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-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-riota 6257  df-om 6701  df-recs 7061  df-rdg 7095  df-r1 8203  df-rank 8204