Theorem hta 9935

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 1928 and weth 10533, using scottexs 9925 to establish the existence of 𝐴.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9934. (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 2179	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
2		scott0s 9926	. . . 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 3003	. . . 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 9925	. . . . 5 ⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
8	3, 7	eqeltri 2835	. . . 4 ⊢ 𝐴 ∈ V
9		hta.2	. . . 4 ⊢ 𝐵 = (℩𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧)
10	8, 9	htalem 9934	. . 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 4077	. . . . 5 ⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ⊆ {𝑥 ∣ 𝜑}
14	3, 13	eqsstri 4030	. . . 4 ⊢ 𝐴 ⊆ {𝑥 ∣ 𝜑}
15	14	sseli 3991	. . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝑥 ∣ 𝜑})
16		df-sbc 3792	. . 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 1535   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712   ≠ wne 2938  ∀wral 3059  Vcvv 3478  [wsbc 3791   ⊆ wss 3963  ∅c0 4339   class class class wbr 5148   We wwe 5640  ‘cfv 6563  ℩crio 7387  rankcrnk 9801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803

This theorem is referenced by: (None)