Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-bdfindis | GIF version |
Description: Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4514 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4514, finds2 4515, finds1 4516. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-bdfindis.bd | ⊢ BOUNDED 𝜑 |
bj-bdfindis.nf0 | ⊢ Ⅎ𝑥𝜓 |
bj-bdfindis.nf1 | ⊢ Ⅎ𝑥𝜒 |
bj-bdfindis.nfsuc | ⊢ Ⅎ𝑥𝜃 |
bj-bdfindis.0 | ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) |
bj-bdfindis.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) |
bj-bdfindis.suc | ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) |
Ref | Expression |
---|---|
bj-bdfindis | ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-bdfindis.nf0 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
2 | 0ex 4055 | . . . 4 ⊢ ∅ ∈ V | |
3 | bj-bdfindis.0 | . . . 4 ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) | |
4 | 1, 2, 3 | elabf2 12989 | . . 3 ⊢ (𝜓 → ∅ ∈ {𝑥 ∣ 𝜑}) |
5 | bj-bdfindis.nf1 | . . . . . 6 ⊢ Ⅎ𝑥𝜒 | |
6 | bj-bdfindis.1 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) | |
7 | 5, 6 | elabf1 12988 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → 𝜒) |
8 | bj-bdfindis.nfsuc | . . . . . 6 ⊢ Ⅎ𝑥𝜃 | |
9 | vex 2689 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
10 | 9 | bj-sucex 13121 | . . . . . 6 ⊢ suc 𝑦 ∈ V |
11 | bj-bdfindis.suc | . . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) | |
12 | 8, 10, 11 | elabf2 12989 | . . . . 5 ⊢ (𝜃 → suc 𝑦 ∈ {𝑥 ∣ 𝜑}) |
13 | 7, 12 | imim12i 59 | . . . 4 ⊢ ((𝜒 → 𝜃) → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) |
14 | 13 | ralimi 2495 | . . 3 ⊢ (∀𝑦 ∈ ω (𝜒 → 𝜃) → ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) |
15 | bj-bdfindis.bd | . . . . 5 ⊢ BOUNDED 𝜑 | |
16 | 15 | bdcab 13047 | . . . 4 ⊢ BOUNDED {𝑥 ∣ 𝜑} |
17 | 16 | bdpeano5 13141 | . . 3 ⊢ ((∅ ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) → ω ⊆ {𝑥 ∣ 𝜑}) |
18 | 4, 14, 17 | syl2an 287 | . 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ω ⊆ {𝑥 ∣ 𝜑}) |
19 | ssabral 3168 | . 2 ⊢ (ω ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥 ∈ ω 𝜑) | |
20 | 18, 19 | sylib 121 | 1 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 Ⅎwnf 1436 ∈ wcel 1480 {cab 2125 ∀wral 2416 ⊆ wss 3071 ∅c0 3363 suc csuc 4287 ωcom 4504 BOUNDED wbd 13010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-nul 4054 ax-pr 4131 ax-un 4355 ax-bd0 13011 ax-bdor 13014 ax-bdex 13017 ax-bdeq 13018 ax-bdel 13019 ax-bdsb 13020 ax-bdsep 13082 ax-infvn 13139 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 df-bdc 13039 df-bj-ind 13125 |
This theorem is referenced by: bj-bdfindisg 13146 bj-bdfindes 13147 bj-nn0suc0 13148 |
Copyright terms: Public domain | W3C validator |