Theorem hta 8760

Description: A ZFC emulation of Hilbert's transfinite axiom. The set B has the properties of Hilbert's epsilon, except that it also depends on a well-ordering R. 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 R We A as an antecedent. Class A collects the sets of the least rank for which ph ( x ) is true. Class B, which emulates the epsilon, is the minimum element in a well-ordering R on A.

If a well-ordering R on A 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 R with a dummy setvar variable, say w, and attach w We A as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point, B (which will have w as a free variable) will no longer be present, and we can eliminate w We A by applying exlimiv 1858 and weth 9317, using scottexs 8750 to establish the existence of A.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 8759. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)

Hypotheses

Ref	Expression
hta.1	|- A = { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) }
hta.2	|- B = ( iota_ z e. A A. w e. A -. w R z )

Assertion

Ref	Expression
hta	|- ( R We A -> ( ph -> [. B / x ]. ph ) )

Distinct variable groups:   x, y   z, w, A   ph, y   w, R, z

Allowed substitution hints:   ph( x, z, w)   A( x, y)   B( x, y, z, w)   R( x, y)

Proof of Theorem hta

Step	Hyp	Ref	Expression
1		19.8a 2052	. . 3 |- ( ph -> E. x ph )
2		scott0s 8751	. . . 4 |- ( E. x ph <-> { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } =/= (/) )
3		hta.1	. . . . 5 |- A = { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) }
4	3	neeq1i 2858	. . . 4 |- ( A =/= (/) <-> { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } =/= (/) )
5	2, 4	bitr4i 267	. . 3 |- ( E. x ph <-> A =/= (/) )
6	1, 5	sylib 208	. 2 |- ( ph -> A =/= (/) )
7		scottexs 8750	. . . . 5 |- { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } e. _V
8	3, 7	eqeltri 2697	. . . 4 |- A e. _V
9		hta.2	. . . 4 |- B = ( iota_ z e. A A. w e. A -. w R z )
10	8, 9	htalem 8759	. . 3 |- ( ( R We A /\ A =/= (/) ) -> B e. A )
11	10	ex 450	. 2 |- ( R We A -> ( A =/= (/) -> B e. A ) )
12		simpl 473	. . . . . 6 |- ( ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) -> ph )
13	12	ss2abi 3674	. . . . 5 |- { x | ( ph /\ A. y ( [. y / x ]. ph -> ( rank ` x ) C_ ( rank ` y ) ) ) } C_ { x | ph }
14	3, 13	eqsstri 3635	. . . 4 |- A C_ { x | ph }
15	14	sseli 3599	. . 3 |- ( B e. A -> B e. { x | ph } )
16		df-sbc 3436	. . 3 |- ( [. B / x ]. ph <-> B e. { x | ph } )
17	15, 16	sylibr 224	. 2 |- ( B e. A -> [. B / x ]. ph )
18	6, 11, 17	syl56 36	1 |- ( R We A -> ( ph -> [. B / x ]. ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   _Vcvv 3200   [.wsbc 3435   C_ wss 3574   (/)c0 3915   class class class wbr 4653   We wwe 5072   ` cfv 5888   iota_crio 6610   rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by: (None)