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Mirrors > Home > MPE Home > Th. List > hta | Structured version Visualization version GIF version |
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 1922 and weth 9905, using scottexs 9304 to establish the existence of 𝐴. For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9313. (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 2170 | . . 3 ⊢ (𝜑 → ∃𝑥𝜑) | |
2 | scott0s 9305 | . . . 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 3077 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅) |
5 | 2, 4 | bitr4i 279 | . . 3 ⊢ (∃𝑥𝜑 ↔ 𝐴 ≠ ∅) |
6 | 1, 5 | sylib 219 | . 2 ⊢ (𝜑 → 𝐴 ≠ ∅) |
7 | scottexs 9304 | . . . . 5 ⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V | |
8 | 3, 7 | eqeltri 2906 | . . . 4 ⊢ 𝐴 ∈ V |
9 | hta.2 | . . . 4 ⊢ 𝐵 = (℩𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧) | |
10 | 8, 9 | htalem 9313 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴) |
11 | 10 | ex 413 | . 2 ⊢ (𝑅 We 𝐴 → (𝐴 ≠ ∅ → 𝐵 ∈ 𝐴)) |
12 | simpl 483 | . . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦))) → 𝜑) | |
13 | 12 | ss2abi 4040 | . . . . 5 ⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ⊆ {𝑥 ∣ 𝜑} |
14 | 3, 13 | eqsstri 3998 | . . . 4 ⊢ 𝐴 ⊆ {𝑥 ∣ 𝜑} |
15 | 14 | sseli 3960 | . . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝑥 ∣ 𝜑}) |
16 | df-sbc 3770 | . . 3 ⊢ ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}) | |
17 | 15, 16 | sylibr 235 | . 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 396 ∀wal 1526 = wceq 1528 ∃wex 1771 ∈ wcel 2105 {cab 2796 ≠ wne 3013 ∀wral 3135 Vcvv 3492 [wsbc 3769 ⊆ wss 3933 ∅c0 4288 class class class wbr 5057 We wwe 5506 ‘cfv 6348 ℩crio 7102 rankcrnk 9180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-reg 9044 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-r1 9181 df-rank 9182 |
This theorem is referenced by: (None) |
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