Theorem hta 9328

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 1931 and weth 9919, using scottexs 9318 to establish the existence of 𝐴.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9327. (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 9319	. . . 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 3082	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
5	2, 4	bitr4i 280	. . 3 ⊢ (∃𝑥𝜑 ↔ 𝐴 ≠ ∅)
6	1, 5	sylib 220	. 2 ⊢ (𝜑 → 𝐴 ≠ ∅)
7		scottexs 9318	. . . . 5 ⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
8	3, 7	eqeltri 2911	. . . 4 ⊢ 𝐴 ∈ V
9		hta.2	. . . 4 ⊢ 𝐵 = (℩𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧)
10	8, 9	htalem 9327	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴)
11	10	ex 415	. 2 ⊢ (𝑅 We 𝐴 → (𝐴 ≠ ∅ → 𝐵 ∈ 𝐴))
12		simpl 485	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦))) → 𝜑)
13	12	ss2abi 4045	. . . . 5 ⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ⊆ {𝑥 ∣ 𝜑}
14	3, 13	eqsstri 4003	. . . 4 ⊢ 𝐴 ⊆ {𝑥 ∣ 𝜑}
15	14	sseli 3965	. . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝑥 ∣ 𝜑})
16		df-sbc 3775	. . 3 ⊢ ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})
17	15, 16	sylibr 236	. 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 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2801   ≠ wne 3018  ∀wral 3140  Vcvv 3496  [wsbc 3774   ⊆ wss 3938  ∅c0 4293   class class class wbr 5068   We wwe 5515  ‘cfv 6357  ℩crio 7115  rankcrnk 9194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-reg 9058  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-r1 9195  df-rank 9196

This theorem is referenced by: (None)