![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > on2ind | Structured version Visualization version GIF version |
Description: Double induction over ordinal numbers. (Contributed by Scott Fenton, 26-Aug-2024.) |
Ref | Expression |
---|---|
on2ind.1 | ⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓)) |
on2ind.2 | ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒)) |
on2ind.3 | ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒)) |
on2ind.4 | ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏)) |
on2ind.5 | ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂)) |
on2ind.i | ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒 ∧ ∀𝑐 ∈ 𝑎 𝜓 ∧ ∀𝑑 ∈ 𝑏 𝜃) → 𝜑)) |
Ref | Expression |
---|---|
on2ind | ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onfr 6404 | . 2 ⊢ E Fr On | |
2 | epweon 7762 | . . 3 ⊢ E We On | |
3 | weso 5668 | . . 3 ⊢ ( E We On → E Or On) | |
4 | sopo 5608 | . . 3 ⊢ ( E Or On → E Po On) | |
5 | 2, 3, 4 | mp2b 10 | . 2 ⊢ E Po On |
6 | epse 5660 | . 2 ⊢ E Se On | |
7 | on2ind.1 | . 2 ⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓)) | |
8 | on2ind.2 | . 2 ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒)) | |
9 | on2ind.3 | . 2 ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒)) | |
10 | on2ind.4 | . 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏)) | |
11 | on2ind.5 | . 2 ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂)) | |
12 | predon 7773 | . . . . . 6 ⊢ (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎) | |
13 | 12 | adantr 482 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑎) = 𝑎) |
14 | predon 7773 | . . . . . . 7 ⊢ (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏) | |
15 | 14 | adantl 483 | . . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑏) = 𝑏) |
16 | 15 | raleqdv 3326 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑 ∈ 𝑏 𝜒)) |
17 | 13, 16 | raleqbidv 3343 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒)) |
18 | 13 | raleqdv 3326 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑐 ∈ 𝑎 𝜓)) |
19 | 15 | raleqdv 3326 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃 ↔ ∀𝑑 ∈ 𝑏 𝜃)) |
20 | 17, 18, 19 | 3anbi123d 1437 | . . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃) ↔ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒 ∧ ∀𝑐 ∈ 𝑎 𝜓 ∧ ∀𝑑 ∈ 𝑏 𝜃))) |
21 | on2ind.i | . . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒 ∧ ∀𝑐 ∈ 𝑎 𝜓 ∧ ∀𝑑 ∈ 𝑏 𝜃) → 𝜑)) | |
22 | 20, 21 | sylbid 239 | . 2 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃) → 𝜑)) |
23 | 1, 5, 6, 1, 5, 6, 7, 8, 9, 10, 11, 22 | xpord2ind 8134 | 1 ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 E cep 5580 Po wpo 5587 Or wor 5588 We wwe 5631 Predcpred 6300 Oncon0 6365 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-iota 6496 df-fun 6546 df-fv 6552 df-1st 7975 df-2nd 7976 |
This theorem is referenced by: naddcllem 8675 naddcom 8681 naddsuc2 42143 naddgeoa 42145 |
Copyright terms: Public domain | W3C validator |