![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > finds2 | Structured version Visualization version 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 5303 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | finds2.1 | . . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
4 | 3 | imbi2d 341 | . . . . . 6 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓))) |
5 | 2, 4 | elab 3666 | . . . . 5 ⊢ (∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜓)) |
6 | 1, 5 | mpbir 230 | . . . 4 ⊢ ∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} |
7 | finds2.5 | . . . . . . 7 ⊢ (𝑦 ∈ ω → (𝜏 → (𝜒 → 𝜃))) | |
8 | 7 | a2d 29 | . . . . . 6 ⊢ (𝑦 ∈ ω → ((𝜏 → 𝜒) → (𝜏 → 𝜃))) |
9 | vex 3479 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
10 | finds2.2 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
11 | 10 | imbi2d 341 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜒))) |
12 | 9, 11 | elab 3666 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜒)) |
13 | 9 | sucex 7781 | . . . . . . 7 ⊢ suc 𝑦 ∈ V |
14 | finds2.3 | . . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
15 | 14 | imbi2d 341 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜃))) |
16 | 13, 15 | elab 3666 | . . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜃)) |
17 | 8, 12, 16 | 3imtr4g 296 | . . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})) |
18 | 17 | rgen 3064 | . . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)}) |
19 | peano5 7871 | . . . 4 ⊢ ((∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)}) | |
20 | 6, 18, 19 | mp2an 691 | . . 3 ⊢ ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)} |
21 | 20 | sseli 3976 | . 2 ⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)}) |
22 | abid 2714 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜑)) | |
23 | 21, 22 | sylib 217 | 1 ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 ⊆ wss 3946 ∅c0 4320 suc csuc 6358 ωcom 7842 |
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-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 ax-un 7712 |
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 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-tr 5262 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-om 7843 |
This theorem is referenced by: finds1 7879 onnseq 8331 nnacl 8599 nnmcl 8600 nnecl 8601 nnacom 8605 nnaass 8610 nndi 8611 nnmass 8612 nnmsucr 8613 nnmcom 8614 nnmordi 8619 omsmolem 8644 isinf 9248 isinfOLD 9249 unblem2 9284 fiint 9312 dffi3 9413 card2inf 9537 cantnfle 9653 cantnflt 9654 cantnflem1 9671 cnfcom 9682 trcl 9710 fseqenlem1 10006 nnadju 10179 infpssrlem3 10287 fin23lem26 10307 axdc3lem2 10433 axdc4lem 10437 axdclem2 10502 wunr1om 10701 wuncval2 10729 tskr1om 10749 grothomex 10811 peano5nni 12202 precsexlem6 27625 precsexlem7 27626 neibastop2lem 35150 finxpreclem6 36182 domalom 36190 oaabsb 41915 |
Copyright terms: Public domain | W3C validator |