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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 6394 | . 2 ⊢ E Fr On | |
2 | epweon 7756 | . . 3 ⊢ E We On | |
3 | weso 5658 | . . 3 ⊢ ( E We On → E Or On) | |
4 | sopo 5598 | . . 3 ⊢ ( E Or On → E Po On) | |
5 | 2, 3, 4 | mp2b 10 | . 2 ⊢ E Po On |
6 | epse 5650 | . 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 7767 | . . . . . 6 ⊢ (𝑎 ∈ On → Pred( E , On, 𝑎) = 𝑎) | |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → Pred( E , On, 𝑎) = 𝑎) |
14 | predon 7767 | . . . . . . 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 3317 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑑 ∈ 𝑏 𝜒)) |
17 | 13, 16 | raleqbidv 3334 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)∀𝑑 ∈ Pred ( E , On, 𝑏)𝜒 ↔ ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒)) |
18 | 13 | raleqdv 3317 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑐 ∈ Pred ( E , On, 𝑎)𝜓 ↔ ∀𝑐 ∈ 𝑎 𝜓)) |
19 | 15 | raleqdv 3317 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ Pred ( E , On, 𝑏)𝜃 ↔ ∀𝑑 ∈ 𝑏 𝜃)) |
20 | 17, 18, 19 | 3anbi123d 1432 | . . 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 8129 | 1 ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 E cep 5570 Po wpo 5577 Or wor 5578 We wwe 5621 Predcpred 6290 Oncon0 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-iota 6486 df-fun 6536 df-fv 6542 df-1st 7969 df-2nd 7970 |
This theorem is referenced by: naddcllem 8672 naddcom 8678 naddsuc2 42693 naddgeoa 42695 |
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