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Theorem hta 9314
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 https://us.metamath.org/downloads/choice.txt (copy of obsolete link http://ghilbert.org/choice.txt) and https://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 𝜑(𝑥) is true. Class 𝐵, which emulates Hilbert's 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 1922 and weth 9905, using scottexs 9304 to establish the existence of 𝐴.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9313. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)

Hypotheses
Ref Expression
hta.1 𝐴 = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
hta.2 𝐵 = (𝑧𝐴𝑤𝐴 ¬ 𝑤𝑅𝑧)
Assertion
Ref Expression
hta (𝑅 We 𝐴 → (𝜑[𝐵 / 𝑥]𝜑))
Distinct variable groups:   𝑥,𝑦   𝑧,𝑤,𝐴   𝜑,𝑦   𝑤,𝑅,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑧,𝑤)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑧,𝑤)   𝑅(𝑥,𝑦)

Proof of Theorem hta
StepHypRef Expression
1 19.8a 2170 . . 3 (𝜑 → ∃𝑥𝜑)
2 scott0s 9305 . . . 4 (∃𝑥𝜑 ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
3 hta.1 . . . . 5 𝐴 = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
43neeq1i 3077 . . . 4 (𝐴 ≠ ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
52, 4bitr4i 279 . . 3 (∃𝑥𝜑𝐴 ≠ ∅)
61, 5sylib 219 . 2 (𝜑𝐴 ≠ ∅)
7 scottexs 9304 . . . . 5 {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
83, 7eqeltri 2906 . . . 4 𝐴 ∈ V
9 hta.2 . . . 4 𝐵 = (𝑧𝐴𝑤𝐴 ¬ 𝑤𝑅𝑧)
108, 9htalem 9313 . . 3 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐵𝐴)
1110ex 413 . 2 (𝑅 We 𝐴 → (𝐴 ≠ ∅ → 𝐵𝐴))
12 simpl 483 . . . . . 6 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦))) → 𝜑)
1312ss2abi 4040 . . . . 5 {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ⊆ {𝑥𝜑}
143, 13eqsstri 3998 . . . 4 𝐴 ⊆ {𝑥𝜑}
1514sseli 3960 . . 3 (𝐵𝐴𝐵 ∈ {𝑥𝜑})
16 df-sbc 3770 . . 3 ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑})
1715, 16sylibr 235 . 2 (𝐵𝐴[𝐵 / 𝑥]𝜑)
186, 11, 17syl56 36 1 (𝑅 We 𝐴 → (𝜑[𝐵 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1526   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wne 3013  wral 3135  Vcvv 3492  [wsbc 3769  wss 3933  c0 4288   class class class wbr 5057   We wwe 5506  cfv 6348  crio 7102  rankcrnk 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-r1 9181  df-rank 9182
This theorem is referenced by: (None)
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