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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 5308 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | finds.1 | . . . . . 6 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | elab 3669 | . . . . 5 ⊢ (∅ ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
5 | 1, 4 | mpbir 230 | . . . 4 ⊢ ∅ ∈ {𝑥 ∣ 𝜑} |
6 | finds.6 | . . . . . 6 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃)) | |
7 | vex 3479 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
8 | finds.2 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
9 | 7, 8 | elab 3669 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒) |
10 | 7 | sucex 7794 | . . . . . . 7 ⊢ suc 𝑦 ∈ V |
11 | finds.3 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
12 | 10, 11 | elab 3669 | . . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜃) |
13 | 6, 9, 12 | 3imtr4g 296 | . . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) |
14 | 13 | rgen 3064 | . . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}) |
15 | peano5 7884 | . . . 4 ⊢ ((∅ ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) → ω ⊆ {𝑥 ∣ 𝜑}) | |
16 | 5, 14, 15 | mp2an 691 | . . 3 ⊢ ω ⊆ {𝑥 ∣ 𝜑} |
17 | 16 | sseli 3979 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
18 | finds.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
19 | 18 | elabg 3667 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜏)) |
20 | 17, 19 | mpbid 231 | 1 ⊢ (𝐴 ∈ ω → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 ⊆ wss 3949 ∅c0 4323 suc csuc 6367 ωcom 7855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-om 7856 |
This theorem is referenced by: findsg 7890 findes 7893 seqomlem1 8450 nna0r 8609 nnm0r 8610 nnawordi 8621 nneob 8655 enrefnn 9047 pssnn 9168 nneneq 9209 nneneqOLD 9221 pssnnOLD 9265 inf3lem1 9623 inf3lem2 9624 cantnfval2 9664 cantnfp1lem3 9675 ttrclss 9715 ttrclselem2 9721 r1fin 9768 ackbij1lem14 10228 ackbij1lem16 10230 ackbij1 10233 ackbij2lem2 10235 ackbij2lem3 10236 infpssrlem4 10301 fin23lem14 10328 fin23lem34 10341 itunitc1 10415 ituniiun 10417 om2uzuzi 13914 om2uzlti 13915 om2uzrdg 13921 uzrdgxfr 13932 hashgadd 14337 mreexexd 17592 precsexlem8 27660 precsexlem9 27661 satfrel 34358 satfdm 34360 satfrnmapom 34361 satf0op 34368 satf0n0 34369 sat1el2xp 34370 fmlafvel 34376 fmlaomn0 34381 gonar 34386 goalr 34388 satffun 34400 findfvcl 35337 finxp00 36283 onmcl 42081 naddonnn 42146 |
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