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Theorem hta 7583
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 set 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 1624 and weth 8138, using scottexs 7573 to establish the existence of 
A.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 7582. (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
StepHypRef Expression
1 19.8a 1730 . . 3  |-  ( ph  ->  E. x ph )
2 scott0s 7574 . . . 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 ) ) ) }
43neeq1i 2469 . . . 4  |-  ( A  =/=  (/)  <->  { x  |  (
ph  /\  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) ) }  =/=  (/) )
52, 4bitr4i 243 . . 3  |-  ( E. x ph  <->  A  =/=  (/) )
61, 5sylib 188 . 2  |-  ( ph  ->  A  =/=  (/) )
7 scottexs 7573 . . . . 5  |-  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) }  e.  _V
83, 7eqeltri 2366 . . . 4  |-  A  e. 
_V
9 hta.2 . . . 4  |-  B  =  ( iota_ z  e.  A A. w  e.  A  -.  w R z )
108, 9htalem 7582 . . 3  |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  B  e.  A )
1110ex 423 . 2  |-  ( R  We  A  ->  ( A  =/=  (/)  ->  B  e.  A ) )
12 simpl 443 . . . . . 6  |-  ( (
ph  /\  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) )  ->  ph )
1312ss2abi 3258 . . . . 5  |-  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) } 
C_  { x  | 
ph }
143, 13eqsstri 3221 . . . 4  |-  A  C_  { x  |  ph }
1514sseli 3189 . . 3  |-  ( B  e.  A  ->  B  e.  { x  |  ph } )
16 df-sbc 3005 . . 3  |-  ( [. B  /  x ]. ph  <->  B  e.  { x  |  ph }
)
1715, 16sylibr 203 . 2  |-  ( B  e.  A  ->  [. B  /  x ]. ph )
186, 11, 17syl56 30 1  |-  ( R  We  A  ->  ( ph  ->  [. B  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   _Vcvv 2801   [.wsbc 3004    C_ wss 3165   (/)c0 3468   class class class wbr 4039    We wwe 4367   ` cfv 5271   iota_crio 6313   rankcrnk 7451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 6320  df-recs 6404  df-rdg 6439  df-r1 7452  df-rank 7453
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