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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 5203 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | finds.1 | . . . . . 6 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | elab 3666 | . . . . 5 ⊢ (∅ ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
5 | 1, 4 | mpbir 233 | . . . 4 ⊢ ∅ ∈ {𝑥 ∣ 𝜑} |
6 | finds.6 | . . . . . 6 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃)) | |
7 | vex 3497 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
8 | finds.2 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
9 | 7, 8 | elab 3666 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒) |
10 | 7 | sucex 7520 | . . . . . . 7 ⊢ suc 𝑦 ∈ V |
11 | finds.3 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
12 | 10, 11 | elab 3666 | . . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜃) |
13 | 6, 9, 12 | 3imtr4g 298 | . . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) |
14 | 13 | rgen 3148 | . . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}) |
15 | peano5 7599 | . . . 4 ⊢ ((∅ ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) → ω ⊆ {𝑥 ∣ 𝜑}) | |
16 | 5, 14, 15 | mp2an 690 | . . 3 ⊢ ω ⊆ {𝑥 ∣ 𝜑} |
17 | 16 | sseli 3962 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
18 | finds.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
19 | 18 | elabg 3665 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜏)) |
20 | 17, 19 | mpbid 234 | 1 ⊢ (𝐴 ∈ ω → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 ⊆ wss 3935 ∅c0 4290 suc csuc 6187 ωcom 7574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-om 7575 |
This theorem is referenced by: findsg 7603 findes 7606 seqomlem1 8080 nna0r 8229 nnm0r 8230 nnawordi 8241 nneob 8273 nneneq 8694 pssnn 8730 inf3lem1 9085 inf3lem2 9086 cantnfval2 9126 cantnfp1lem3 9137 r1fin 9196 ackbij1lem14 9649 ackbij1lem16 9651 ackbij1 9654 ackbij2lem2 9656 ackbij2lem3 9657 infpssrlem4 9722 fin23lem14 9749 fin23lem34 9762 itunitc1 9836 ituniiun 9838 om2uzuzi 13311 om2uzlti 13312 om2uzrdg 13318 uzrdgxfr 13329 hashgadd 13732 mreexexd 16913 satfrel 32609 satfdm 32611 satfrnmapom 32612 satf0op 32619 satf0n0 32620 sat1el2xp 32621 fmlafvel 32627 fmlaomn0 32632 gonar 32637 goalr 32639 satffun 32651 trpredmintr 33065 findfvcl 33795 finxp00 34677 |
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