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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 4129 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | finds2.1 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
4 | 3 | imbi2d 230 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓))) |
5 | 2, 4 | elab 2881 | . . . . 5 ⊢ (∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜓)) |
6 | 1, 5 | mpbir 146 | . . . 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 2740 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
10 | finds2.2 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
11 | 10 | imbi2d 230 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜒))) |
12 | 9, 11 | elab 2881 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜒)) |
13 | 9 | sucex 4497 | . . . . . . 7 ⊢ suc 𝑦 ∈ V |
14 | finds2.3 | . . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
15 | 14 | imbi2d 230 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜃))) |
16 | 13, 15 | elab 2881 | . . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜃)) |
17 | 8, 12, 16 | 3imtr4g 205 | . . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})) |
18 | 17 | rgen 2530 | . . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)}) |
19 | peano5 4596 | . . . 4 ⊢ ((∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)}) | |
20 | 6, 18, 19 | mp2an 426 | . . 3 ⊢ ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)} |
21 | 20 | sseli 3151 | . 2 ⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)}) |
22 | abid 2165 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜑)) | |
23 | 21, 22 | sylib 122 | 1 ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 {cab 2163 ∀wral 2455 ⊆ wss 3129 ∅c0 3422 suc csuc 4364 ωcom 4588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-iinf 4586 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-int 3845 df-suc 4370 df-iom 4589 |
This theorem is referenced by: finds1 4600 frecrdg 6406 nnacl 6478 nnmcl 6479 nnacom 6482 nnaass 6483 nndi 6484 nnmass 6485 nnmsucr 6486 nnmcom 6487 nnsucsssuc 6490 nntri3or 6491 nnaordi 6506 nnaword 6509 nnmordi 6514 nnaordex 6526 fiintim 6925 prarloclem3 7493 frec2uzuzd 10397 frec2uzrdg 10404 |
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