Theorem hta 9938

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 1929 and weth 10536, using scottexs 9928 to establish the existence of 𝐴.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9937. (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 = (℩𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 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

Step	Hyp	Ref	Expression
1		19.8a 2180	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
2		scott0s 9929	. . . 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)))}
4	3	neeq1i 3004	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
5	2, 4	bitr4i 278	. . 3 ⊢ (∃𝑥𝜑 ↔ 𝐴 ≠ ∅)
6	1, 5	sylib 218	. 2 ⊢ (𝜑 → 𝐴 ≠ ∅)
7		scottexs 9928	. . . . 5 ⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
8	3, 7	eqeltri 2836	. . . 4 ⊢ 𝐴 ∈ V
9		hta.2	. . . 4 ⊢ 𝐵 = (℩𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧)
10	8, 9	htalem 9937	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴)
11	10	ex 412	. 2 ⊢ (𝑅 We 𝐴 → (𝐴 ≠ ∅ → 𝐵 ∈ 𝐴))
12		simpl 482	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦))) → 𝜑)
13	12	ss2abi 4066	. . . . 5 ⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ⊆ {𝑥 ∣ 𝜑}
14	3, 13	eqsstri 4029	. . . 4 ⊢ 𝐴 ⊆ {𝑥 ∣ 𝜑}
15	14	sseli 3978	. . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝑥 ∣ 𝜑})
16		df-sbc 3788	. . 3 ⊢ ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})
17	15, 16	sylibr 234	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵 / 𝑥]𝜑)
18	6, 11, 17	syl56 36	1 ⊢ (𝑅 We 𝐴 → (𝜑 → [𝐵 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2713   ≠ wne 2939  ∀wral 3060  Vcvv 3479  [wsbc 3787   ⊆ wss 3950  ∅c0 4332   class class class wbr 5142   We wwe 5635  ‘cfv 6560  ℩crio 7388  rankcrnk 9804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633  ax-inf2 9682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-r1 9805  df-rank 9806

This theorem is referenced by: (None)