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Mirrors > Home > MPE Home > Th. List > tfinds3 | Structured version Visualization version GIF version |
Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.) |
Ref | Expression |
---|---|
tfinds3.1 | ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) |
tfinds3.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
tfinds3.3 | ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) |
tfinds3.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
tfinds3.5 | ⊢ (𝜂 → 𝜓) |
tfinds3.6 | ⊢ (𝑦 ∈ On → (𝜂 → (𝜒 → 𝜃))) |
tfinds3.7 | ⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))) |
Ref | Expression |
---|---|
tfinds3 | ⊢ (𝐴 ∈ On → (𝜂 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfinds3.1 | . . 3 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
2 | 1 | imbi2d 341 | . 2 ⊢ (𝑥 = ∅ → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜓))) |
3 | tfinds3.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
4 | 3 | imbi2d 341 | . 2 ⊢ (𝑥 = 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜒))) |
5 | tfinds3.3 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
6 | 5 | imbi2d 341 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜃))) |
7 | tfinds3.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
8 | 7 | imbi2d 341 | . 2 ⊢ (𝑥 = 𝐴 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜏))) |
9 | tfinds3.5 | . 2 ⊢ (𝜂 → 𝜓) | |
10 | tfinds3.6 | . . 3 ⊢ (𝑦 ∈ On → (𝜂 → (𝜒 → 𝜃))) | |
11 | 10 | a2d 29 | . 2 ⊢ (𝑦 ∈ On → ((𝜂 → 𝜒) → (𝜂 → 𝜃))) |
12 | r19.21v 3177 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) ↔ (𝜂 → ∀𝑦 ∈ 𝑥 𝜒)) | |
13 | tfinds3.7 | . . . 4 ⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))) | |
14 | 13 | a2d 29 | . . 3 ⊢ (Lim 𝑥 → ((𝜂 → ∀𝑦 ∈ 𝑥 𝜒) → (𝜂 → 𝜑))) |
15 | 12, 14 | biimtrid 241 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) → (𝜂 → 𝜑))) |
16 | 2, 4, 6, 8, 9, 11, 15 | tfinds 7801 | 1 ⊢ (𝐴 ∈ On → (𝜂 → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ∅c0 4287 Oncon0 6322 Lim wlim 6323 suc csuc 6324 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
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-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 |
This theorem is referenced by: oacl 8486 omcl 8487 oecl 8488 oawordri 8502 oaass 8513 oarec 8514 omordi 8518 omwordri 8524 odi 8531 omass 8532 oen0 8538 oewordri 8544 oeworde 8545 oeoelem 8550 omabs 8602 tfindsd 42559 |
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