Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4116 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | finds.1 | . . . . . 6 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | elab 2874 | . . . . 5 ⊢ (∅ ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
5 | 1, 4 | mpbir 145 | . . . 4 ⊢ ∅ ∈ {𝑥 ∣ 𝜑} |
6 | finds.6 | . . . . . 6 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃)) | |
7 | vex 2733 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
8 | finds.2 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
9 | 7, 8 | elab 2874 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒) |
10 | 7 | sucex 4483 | . . . . . . 7 ⊢ suc 𝑦 ∈ V |
11 | finds.3 | . . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
12 | 10, 11 | elab 2874 | . . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜃) |
13 | 6, 9, 12 | 3imtr4g 204 | . . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) |
14 | 13 | rgen 2523 | . . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}) |
15 | peano5 4582 | . . . 4 ⊢ ((∅ ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) → ω ⊆ {𝑥 ∣ 𝜑}) | |
16 | 5, 14, 15 | mp2an 424 | . . 3 ⊢ ω ⊆ {𝑥 ∣ 𝜑} |
17 | 16 | sseli 3143 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
18 | finds.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
19 | 18 | elabg 2876 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜏)) |
20 | 17, 19 | mpbid 146 | 1 ⊢ (𝐴 ∈ ω → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1348 ∈ wcel 2141 {cab 2156 ∀wral 2448 ⊆ wss 3121 ∅c0 3414 suc csuc 4350 ωcom 4574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-suc 4356 df-iom 4575 |
This theorem is referenced by: findes 4587 nn0suc 4588 elomssom 4589 ordom 4591 nndceq0 4602 0elnn 4603 omsinds 4606 nna0r 6457 nnm0r 6458 nnsucelsuc 6470 nneneq 6835 php5 6836 php5dom 6841 fidcenumlemrk 6931 fidcenumlemr 6932 nnnninfeq 7104 nnnninfeq2 7105 frec2uzltd 10359 frecuzrdgg 10372 seq3val 10414 seqvalcd 10415 omgadd 10737 zfz1iso 10776 ennnfonelemhom 12370 nninfsellemdc 14043 |
Copyright terms: Public domain | W3C validator |