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Mirrors > Home > MPE Home > Th. List > finds | Structured version Visualization version 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 5026 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | finds.1 | . . . . . 6 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | elab 3558 | . . . . 5 ⊢ (∅ ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
5 | 1, 4 | mpbir 223 | . . . 4 ⊢ ∅ ∈ {𝑥 ∣ 𝜑} |
6 | finds.6 | . . . . . 6 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃)) | |
7 | vex 3401 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
8 | finds.2 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
9 | 7, 8 | elab 3558 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒) |
10 | 7 | sucex 7289 | . . . . . . 7 ⊢ suc 𝑦 ∈ V |
11 | finds.3 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
12 | 10, 11 | elab 3558 | . . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜃) |
13 | 6, 9, 12 | 3imtr4g 288 | . . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) |
14 | 13 | rgen 3104 | . . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}) |
15 | peano5 7367 | . . . 4 ⊢ ((∅ ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) → ω ⊆ {𝑥 ∣ 𝜑}) | |
16 | 5, 14, 15 | mp2an 682 | . . 3 ⊢ ω ⊆ {𝑥 ∣ 𝜑} |
17 | 16 | sseli 3817 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
18 | finds.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
19 | 18 | elabg 3556 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜏)) |
20 | 17, 19 | mpbid 224 | 1 ⊢ (𝐴 ∈ ω → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 {cab 2763 ∀wral 3090 ⊆ wss 3792 ∅c0 4141 suc csuc 5978 ωcom 7343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-tr 4988 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-om 7344 |
This theorem is referenced by: findsg 7371 findes 7374 seqomlem1 7828 nna0r 7973 nnm0r 7974 nnawordi 7985 nneob 8016 nneneq 8431 pssnn 8466 inf3lem1 8822 inf3lem2 8823 cantnfval2 8863 cantnfp1lem3 8874 r1fin 8933 ackbij1lem14 9390 ackbij1lem16 9392 ackbij1 9395 ackbij2lem2 9397 ackbij2lem3 9398 infpssrlem4 9463 fin23lem14 9490 fin23lem34 9503 itunitc1 9577 ituniiun 9579 om2uzuzi 13067 om2uzlti 13068 om2uzrdg 13074 uzrdgxfr 13085 hashgadd 13481 mreexexd 16694 trpredmintr 32319 findfvcl 33034 finxp00 33834 |
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