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Mirrors > Home > ILE Home > Th. List > finds2 | GIF version |
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.) |
Ref | Expression |
---|---|
finds2.1 | ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) |
finds2.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
finds2.3 | ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) |
finds2.4 | ⊢ (𝜏 → 𝜓) |
finds2.5 | ⊢ (𝑦 ∈ ω → (𝜏 → (𝜒 → 𝜃))) |
Ref | Expression |
---|---|
finds2 | ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | finds2.4 | . . . . 5 ⊢ (𝜏 → 𝜓) | |
2 | 0ex 4055 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | finds2.1 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
4 | 3 | imbi2d 229 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓))) |
5 | 2, 4 | elab 2828 | . . . . 5 ⊢ (∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜓)) |
6 | 1, 5 | mpbir 145 | . . . 4 ⊢ ∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} |
7 | finds2.5 | . . . . . . 7 ⊢ (𝑦 ∈ ω → (𝜏 → (𝜒 → 𝜃))) | |
8 | 7 | a2d 26 | . . . . . 6 ⊢ (𝑦 ∈ ω → ((𝜏 → 𝜒) → (𝜏 → 𝜃))) |
9 | vex 2689 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
10 | finds2.2 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
11 | 10 | imbi2d 229 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜒))) |
12 | 9, 11 | elab 2828 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜒)) |
13 | 9 | sucex 4415 | . . . . . . 7 ⊢ suc 𝑦 ∈ V |
14 | finds2.3 | . . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
15 | 14 | imbi2d 229 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜃))) |
16 | 13, 15 | elab 2828 | . . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜃)) |
17 | 8, 12, 16 | 3imtr4g 204 | . . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})) |
18 | 17 | rgen 2485 | . . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)}) |
19 | peano5 4512 | . . . 4 ⊢ ((∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)}) | |
20 | 6, 18, 19 | mp2an 422 | . . 3 ⊢ ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)} |
21 | 20 | sseli 3093 | . 2 ⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)}) |
22 | abid 2127 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜑)) | |
23 | 21, 22 | sylib 121 | 1 ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {cab 2125 ∀wral 2416 ⊆ wss 3071 ∅c0 3363 suc csuc 4287 ωcom 4504 |
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-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 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-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 |
This theorem is referenced by: finds1 4516 frecrdg 6305 nnacl 6376 nnmcl 6377 nnacom 6380 nnaass 6381 nndi 6382 nnmass 6383 nnmsucr 6384 nnmcom 6385 nnsucsssuc 6388 nntri3or 6389 nnaordi 6404 nnaword 6407 nnmordi 6412 nnaordex 6423 fiintim 6817 prarloclem3 7305 frec2uzuzd 10175 frec2uzrdg 10182 |
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