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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dford3 | Structured version Visualization version GIF version |
Description: Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
Ref | Expression |
---|---|
dford3 | ⊢ (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 5955 | . . 3 ⊢ (Ord 𝑁 → Tr 𝑁) | |
2 | ordelord 5963 | . . . . 5 ⊢ ((Ord 𝑁 ∧ 𝑥 ∈ 𝑁) → Ord 𝑥) | |
3 | ordtr 5955 | . . . . 5 ⊢ (Ord 𝑥 → Tr 𝑥) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((Ord 𝑁 ∧ 𝑥 ∈ 𝑁) → Tr 𝑥) |
5 | 4 | ralrimiva 3147 | . . 3 ⊢ (Ord 𝑁 → ∀𝑥 ∈ 𝑁 Tr 𝑥) |
6 | 1, 5 | jca 508 | . 2 ⊢ (Ord 𝑁 → (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥)) |
7 | simpl 475 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → Tr 𝑁) | |
8 | dford3lem1 38378 | . . . . 5 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → ∀𝑎 ∈ 𝑁 (Tr 𝑎 ∧ ∀𝑥 ∈ 𝑎 Tr 𝑥)) | |
9 | dford3lem2 38379 | . . . . . 6 ⊢ ((Tr 𝑎 ∧ ∀𝑥 ∈ 𝑎 Tr 𝑥) → 𝑎 ∈ On) | |
10 | 9 | ralimi 3133 | . . . . 5 ⊢ (∀𝑎 ∈ 𝑁 (Tr 𝑎 ∧ ∀𝑥 ∈ 𝑎 Tr 𝑥) → ∀𝑎 ∈ 𝑁 𝑎 ∈ On) |
11 | 8, 10 | syl 17 | . . . 4 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → ∀𝑎 ∈ 𝑁 𝑎 ∈ On) |
12 | dfss3 3787 | . . . 4 ⊢ (𝑁 ⊆ On ↔ ∀𝑎 ∈ 𝑁 𝑎 ∈ On) | |
13 | 11, 12 | sylibr 226 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → 𝑁 ⊆ On) |
14 | ordon 7217 | . . . 4 ⊢ Ord On | |
15 | 14 | a1i 11 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → Ord On) |
16 | trssord 5958 | . . 3 ⊢ ((Tr 𝑁 ∧ 𝑁 ⊆ On ∧ Ord On) → Ord 𝑁) | |
17 | 7, 13, 15, 16 | syl3anc 1491 | . 2 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → Ord 𝑁) |
18 | 6, 17 | impbii 201 | 1 ⊢ (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 385 ∈ wcel 2157 ∀wral 3089 ⊆ wss 3769 Tr wtr 4945 Ord word 5940 Oncon0 5941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 ax-un 7183 ax-reg 8739 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-tr 4946 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-ord 5944 df-on 5945 df-suc 5947 |
This theorem is referenced by: dford4 38381 |
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