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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 344 | . 2 ⊢ (𝑥 = ∅ → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜓))) |
3 | tfinds3.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
4 | 3 | imbi2d 344 | . 2 ⊢ (𝑥 = 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜒))) |
5 | tfinds3.3 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
6 | 5 | imbi2d 344 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜃))) |
7 | tfinds3.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
8 | 7 | imbi2d 344 | . 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 3088 | . . 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 245 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) → (𝜂 → 𝜑))) |
16 | 2, 4, 6, 8, 9, 11, 15 | tfinds 7616 | 1 ⊢ (𝐴 ∈ On → (𝜂 → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∅c0 4223 Oncon0 6191 Lim wlim 6192 suc csuc 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 |
This theorem is referenced by: oacl 8240 omcl 8241 oecl 8242 oawordri 8256 oaass 8267 oarec 8268 omordi 8272 omwordri 8278 odi 8285 omass 8286 oen0 8292 oewordri 8298 oeworde 8299 oeoelem 8304 omabs 8354 tfindsd 41442 |
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