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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 | eqid 2758 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} | |
2 | onfr 6213 | . 2 ⊢ E Fr On | |
3 | epweon 7502 | . . 3 ⊢ E We On | |
4 | weso 5519 | . . 3 ⊢ ( E We On → E Or On) | |
5 | sopo 5465 | . . 3 ⊢ ( E Or On → E Po On) | |
6 | 3, 4, 5 | mp2b 10 | . 2 ⊢ E Po On |
7 | epse 5511 | . 2 ⊢ E Se On | |
8 | on2ind.1 | . 2 ⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓)) | |
9 | on2ind.2 | . 2 ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒)) | |
10 | on2ind.3 | . 2 ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒)) | |
11 | on2ind.4 | . 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏)) | |
12 | on2ind.5 | . 2 ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂)) | |
13 | predon 7511 | . . . . . 6 ⊢ (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎) | |
14 | 13 | adantr 484 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑎) = 𝑎) |
15 | predon 7511 | . . . . . . 7 ⊢ (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏) | |
16 | 15 | adantl 485 | . . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑏) = 𝑏) |
17 | 16 | raleqdv 3329 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑 ∈ 𝑏 𝜒)) |
18 | 14, 17 | raleqbidv 3319 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒)) |
19 | 14 | raleqdv 3329 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑐 ∈ 𝑎 𝜓)) |
20 | 16 | raleqdv 3329 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃 ↔ ∀𝑑 ∈ 𝑏 𝜃)) |
21 | 18, 19, 20 | 3anbi123d 1433 | . . 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))) |
22 | on2ind.i | . . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒 ∧ ∀𝑐 ∈ 𝑎 𝜓 ∧ ∀𝑑 ∈ 𝑏 𝜃) → 𝜑)) | |
23 | 21, 22 | sylbid 243 | . 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)) |
24 | 1, 2, 6, 7, 2, 6, 7, 8, 9, 10, 11, 12, 23 | xpord2ind 33362 | 1 ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ∀wral 3070 class class class wbr 5036 {copab 5098 E cep 5438 Po wpo 5445 Or wor 5446 We wwe 5486 × cxp 5526 Predcpred 6130 Oncon0 6174 ‘cfv 6340 1st c1st 7697 2nd c2nd 7698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-se 5488 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-iota 6299 df-fun 6342 df-fv 6348 df-1st 7699 df-2nd 7700 |
This theorem is referenced by: naddcllem 33429 naddcom 33433 |
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