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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 4103 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | finds2.1 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
4 | 3 | imbi2d 229 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓))) |
5 | 2, 4 | elab 2865 | . . . . 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 2724 | . . . . . . 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 2865 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜒)) |
13 | 9 | sucex 4470 | . . . . . . 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 2865 | . . . . . 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 2517 | . . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)}) |
19 | peano5 4569 | . . . 4 ⊢ ((∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)}) | |
20 | 6, 18, 19 | mp2an 423 | . . 3 ⊢ ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)} |
21 | 20 | sseli 3133 | . 2 ⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)}) |
22 | abid 2152 | . 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 1342 ∈ wcel 2135 {cab 2150 ∀wral 2442 ⊆ wss 3111 ∅c0 3404 suc csuc 4337 ωcom 4561 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-suc 4343 df-iom 4562 |
This theorem is referenced by: finds1 4573 frecrdg 6367 nnacl 6439 nnmcl 6440 nnacom 6443 nnaass 6444 nndi 6445 nnmass 6446 nnmsucr 6447 nnmcom 6448 nnsucsssuc 6451 nntri3or 6452 nnaordi 6467 nnaword 6470 nnmordi 6475 nnaordex 6486 fiintim 6885 prarloclem3 7429 frec2uzuzd 10327 frec2uzrdg 10334 |
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