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| 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.) | 
| Ref | Expression | 
|---|---|
| hta.1 | ⊢ 𝐴 = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} | 
| hta.2 | ⊢ 𝐵 = (℩𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧) | 
| Ref | Expression | 
|---|---|
| hta | ⊢ (𝑅 We 𝐴 → (𝜑 → [𝐵 / 𝑥]𝜑)) | 
| 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)) | 
| Colors of variables: wff setvar class | 
| 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) | 
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