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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 6408 | . 2 ⊢ E Fr On | |
2 | epweon 7777 | . . 3 ⊢ E We On | |
3 | weso 5669 | . . 3 ⊢ ( E We On → E Or On) | |
4 | sopo 5609 | . . 3 ⊢ ( E Or On → E Po On) | |
5 | 2, 3, 4 | mp2b 10 | . 2 ⊢ E Po On |
6 | epse 5661 | . 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 7788 | . . . . . 6 ⊢ (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎) | |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑎) = 𝑎) |
14 | predon 7788 | . . . . . . 7 ⊢ (𝑏 ∈ On → Pred( E , On, 𝑏) = 𝑏) | |
15 | 14 | adantl 481 | . . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑏) = 𝑏) |
16 | 15 | raleqdv 3322 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑 ∈ 𝑏 𝜒)) |
17 | 13, 16 | raleqbidv 3339 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒)) |
18 | 13 | raleqdv 3322 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑐 ∈ 𝑎 𝜓)) |
19 | 15 | raleqdv 3322 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃 ↔ ∀𝑑 ∈ 𝑏 𝜃)) |
20 | 17, 18, 19 | 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))) |
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 8153 | 1 ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3058 E cep 5581 Po wpo 5588 Or wor 5589 We wwe 5632 Predcpred 6304 Oncon0 6369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-iota 6500 df-fun 6550 df-fv 6556 df-1st 7993 df-2nd 7994 |
This theorem is referenced by: naddcllem 8697 naddcom 8703 naddsuc2 42822 naddgeoa 42824 |
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