Theorem hta 9955

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 1963 and weth 10566, using scottex 9926 to establish the existence of 𝐴.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9954. (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 2218	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
2		scott0bs 9937	. . . 4 ⊢ (∃𝑥𝜑 ↔ Scott {𝑥 ∣ 𝜑} ≠ ∅)
3		hta.1	. . . . 5 ⊢ 𝐴 = Scott {𝑥 ∣ 𝜑}
4	3	neeq1i 3020	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ Scott {𝑥 ∣ 𝜑} ≠ ∅)
5	2, 4	bitr4i 281	. . 3 ⊢ (∃𝑥𝜑 ↔ 𝐴 ≠ ∅)
6	1, 5	sylib 221	. 2 ⊢ (𝜑 → 𝐴 ≠ ∅)
7		scottex 9926	. . . . 5 ⊢ Scott {𝑥 ∣ 𝜑} ∈ V
8	3, 7	eqeltri 2857	. . . 4 ⊢ 𝐴 ∈ V
9		hta.2	. . . 4 ⊢ 𝐵 = (℩𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑧𝑅𝑦)
10	8, 9	htalem 9954	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴)
11	10	ex 418	. 2 ⊢ (𝑅 We 𝐴 → (𝐴 ≠ ∅ → 𝐵 ∈ 𝐴))
12		scottss 9928	. . . . 5 ⊢ Scott {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑}
13	3, 12	eqsstri 3977	. . . 4 ⊢ 𝐴 ⊆ {𝑥 ∣ 𝜑}
14	13	sseli 3927	. . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝑥 ∣ 𝜑})
15		df-sbc 3740	. . 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 2145  {cab 2739   ≠ wne 2956  ∀wral 3077  Vcvv 3451  [wsbc 3739  ∅c0 4279   class class class wbr 5103   We wwe 5603  ℩crio 7374  Scott cscott 9921

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-reg 9579  ax-inf2 9635

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-r1 9761  df-rank 9762  df-scott 9922

This theorem is used by: (None)