Theorem hta 9999

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 1965 and weth 10612, using scottex 9970 to establish the existence of 𝐴.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9998. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.) Use the Scott operation. (Revised by BTernaryTau, 22-Jul-2026.)

Hypotheses

Ref	Expression
hta.1	⊢ 𝐴 = Scott {𝑥 ∣ 𝜑}
hta.2	⊢ 𝐵 = (℩𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑧𝑅𝑦)

Assertion

Ref	Expression
hta	⊢ (𝑅 We 𝐴 → (𝜑 → [𝐵 / 𝑥]𝜑))

Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝑅,𝑧

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

Proof of Theorem hta

Step	Hyp	Ref	Expression
1		19.8a 2221	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
2		scott0bs 9981	. . . 4 ⊢ (∃𝑥𝜑 ↔ Scott {𝑥 ∣ 𝜑} ≠ ∅)
3		hta.1	. . . . 5 ⊢ 𝐴 = Scott {𝑥 ∣ 𝜑}
4	3	neeq1i 3023	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ Scott {𝑥 ∣ 𝜑} ≠ ∅)
5	2, 4	bitr4i 281	. . 3 ⊢ (∃𝑥𝜑 ↔ 𝐴 ≠ ∅)
6	1, 5	sylib 221	. 2 ⊢ (𝜑 → 𝐴 ≠ ∅)
7		scottex 9970	. . . . 5 ⊢ Scott {𝑥 ∣ 𝜑} ∈ V
8	3, 7	eqeltri 2860	. . . 4 ⊢ 𝐴 ∈ V
9		hta.2	. . . 4 ⊢ 𝐵 = (℩𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑧𝑅𝑦)
10	8, 9	htalem 9998	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴)
11	10	ex 418	. 2 ⊢ (𝑅 We 𝐴 → (𝐴 ≠ ∅ → 𝐵 ∈ 𝐴))
12		scottss 9972	. . . . 5 ⊢ Scott {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
13	3, 12	eqsstri 3959	. . . 4 ⊢ 𝐴 ⊆ {𝑥 ∣ 𝜑}
14	13	sseli 3909	. . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝑥 ∣ 𝜑})
15		df-sbc 3743	. . 3 ⊢ ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})
16	14, 15	sylibr 237	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵 / 𝑥]𝜑)
17	6, 11, 16	syl56 37	1 ⊢ (𝑅 We 𝐴 → (𝜑 → [𝐵 / 𝑥]𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  Vcvv 3454  [wsbc 3742  ∅c0 4281   class class class wbr 5108   We wwe 5610  ℩crio 7407  Scott cscott 9965

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-reg 9623  ax-inf2 9679

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-r1 9805  df-rank 9806  df-scott 9966

This theorem is used by: (None)