![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 340 | . 2 ⊢ (𝑥 = ∅ → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜓))) |
3 | tfinds3.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
4 | 3 | imbi2d 340 | . 2 ⊢ (𝑥 = 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜒))) |
5 | tfinds3.3 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
6 | 5 | imbi2d 340 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜃))) |
7 | tfinds3.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
8 | 7 | imbi2d 340 | . 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 242 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) → (𝜂 → 𝜑))) |
16 | 2, 4, 6, 8, 9, 11, 15 | tfinds 7880 | 1 ⊢ (𝐴 ∈ On → (𝜂 → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∅c0 4338 Oncon0 6385 Lim wlim 6386 suc csuc 6387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 |
This theorem is referenced by: oacl 8571 omcl 8572 oecl 8573 oawordri 8586 oaass 8597 oarec 8598 omordi 8602 omwordri 8608 odi 8615 omass 8616 oen0 8622 oewordri 8628 oeworde 8629 oeoelem 8634 omabs 8687 tfindsd 44200 |
Copyright terms: Public domain | W3C validator |