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Theorem hta 9302
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 1932 and weth 9894, using scottexs 9292 to establish the existence of 𝐴.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9301. (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 2181 . . 3 (𝜑 → ∃𝑥𝜑)
2 scott0s 9293 . . . 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 3071 . . . 4 (𝐴 ≠ ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
52, 4bitr4i 281 . . 3 (∃𝑥𝜑𝐴 ≠ ∅)
61, 5sylib 221 . 2 (𝜑𝐴 ≠ ∅)
7 scottexs 9292 . . . . 5 {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
83, 7eqeltri 2908 . . . 4 𝐴 ∈ V
9 hta.2 . . . 4 𝐵 = (𝑧𝐴𝑤𝐴 ¬ 𝑤𝑅𝑧)
108, 9htalem 9301 . . 3 ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐵𝐴)
1110ex 416 . 2 (𝑅 We 𝐴 → (𝐴 ≠ ∅ → 𝐵𝐴))
12 simpl 486 . . . . . 6 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦))) → 𝜑)
1312ss2abi 4019 . . . . 5 {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ⊆ {𝑥𝜑}
143, 13eqsstri 3977 . . . 4 𝐴 ⊆ {𝑥𝜑}
1514sseli 3939 . . 3 (𝐵𝐴𝐵 ∈ {𝑥𝜑})
16 df-sbc 3750 . . 3 ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑})
1715, 16sylibr 237 . 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 399  wal 1536   = wceq 1538  wex 1781  wcel 2115  {cab 2799  wne 3007  wral 3126  Vcvv 3471  [wsbc 3749  wss 3910  c0 4266   class class class wbr 5039   We wwe 5486  cfv 6328  crio 7087  rankcrnk 9168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-reg 9032  ax-inf2 9080
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-r1 9169  df-rank 9170
This theorem is referenced by: (None)
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