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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 6425 | . 2 ⊢ E Fr On | |
2 | epweon 7794 | . . 3 ⊢ E We On | |
3 | weso 5680 | . . 3 ⊢ ( E We On → E Or On) | |
4 | sopo 5616 | . . 3 ⊢ ( E Or On → E Po On) | |
5 | 2, 3, 4 | mp2b 10 | . 2 ⊢ E Po On |
6 | epse 5671 | . 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 7805 | . . . . . 6 ⊢ (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎) | |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑎) = 𝑎) |
14 | predon 7805 | . . . . . . 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 3324 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑 ∈ 𝑏 𝜒)) |
17 | 13, 16 | raleqbidv 3344 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒)) |
18 | 13 | raleqdv 3324 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑐 ∈ 𝑎 𝜓)) |
19 | 15 | raleqdv 3324 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃 ↔ ∀𝑑 ∈ 𝑏 𝜃)) |
20 | 17, 18, 19 | 3anbi123d 1435 | . . 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 240 | . 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 8172 | 1 ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 E cep 5588 Po wpo 5595 Or wor 5596 We wwe 5640 Predcpred 6322 Oncon0 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: naddcllem 8713 naddcom 8719 naddsuc2 8738 naddgeoa 43384 |
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