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Mirrors > Home > ILE Home > Th. List > finds | GIF version |
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.) |
Ref | Expression |
---|---|
finds.1 | ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) |
finds.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
finds.3 | ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) |
finds.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
finds.5 | ⊢ 𝜓 |
finds.6 | ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
finds | ⊢ (𝐴 ∈ ω → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | finds.5 | . . . . 5 ⊢ 𝜓 | |
2 | 0ex 3905 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | finds.1 | . . . . . 6 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | elab 2738 | . . . . 5 ⊢ (∅ ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
5 | 1, 4 | mpbir 144 | . . . 4 ⊢ ∅ ∈ {𝑥 ∣ 𝜑} |
6 | finds.6 | . . . . . 6 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃)) | |
7 | vex 2604 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
8 | finds.2 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
9 | 7, 8 | elab 2738 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒) |
10 | 7 | sucex 4243 | . . . . . . 7 ⊢ suc 𝑦 ∈ V |
11 | finds.3 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
12 | 10, 11 | elab 2738 | . . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜃) |
13 | 6, 9, 12 | 3imtr4g 203 | . . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) |
14 | 13 | rgen 2416 | . . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}) |
15 | peano5 4339 | . . . 4 ⊢ ((∅ ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) → ω ⊆ {𝑥 ∣ 𝜑}) | |
16 | 5, 14, 15 | mp2an 416 | . . 3 ⊢ ω ⊆ {𝑥 ∣ 𝜑} |
17 | 16 | sseli 2995 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
18 | finds.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
19 | 18 | elabg 2739 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜏)) |
20 | 17, 19 | mpbid 145 | 1 ⊢ (𝐴 ∈ ω → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 {cab 2067 ∀wral 2348 ⊆ wss 2973 ∅c0 3251 suc csuc 4120 ωcom 4331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-suc 4126 df-iom 4332 |
This theorem is referenced by: findes 4344 nn0suc 4345 elnn 4346 ordom 4347 nndceq0 4357 0elnn 4358 nna0r 6080 nnm0r 6081 nnsucelsuc 6093 nneneq 6343 php5 6344 php5dom 6349 frec2uzltd 9405 |
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