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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 342 | . 2 ⊢ (𝑥 = ∅ → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜓))) |
3 | tfinds3.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
4 | 3 | imbi2d 342 | . 2 ⊢ (𝑥 = 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜒))) |
5 | tfinds3.3 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
6 | 5 | imbi2d 342 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜃))) |
7 | tfinds3.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
8 | 7 | imbi2d 342 | . 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 3172 | . . 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 | syl5bi 243 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) → (𝜂 → 𝜑))) |
16 | 2, 4, 6, 8, 9, 11, 15 | tfinds 7563 | 1 ⊢ (𝐴 ∈ On → (𝜂 → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∅c0 4288 Oncon0 6184 Lim wlim 6185 suc csuc 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 |
This theorem is referenced by: oacl 8149 omcl 8150 oecl 8151 oawordri 8165 oaass 8176 oarec 8177 omordi 8181 omwordri 8187 odi 8194 omass 8195 oen0 8201 oewordri 8207 oeworde 8208 oeoelem 8213 omabs 8263 tfindsd 40441 |
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