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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. Definition 1.2 of [Schloeder] p. 1. (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 6383 | . . 3 ⊢ (Ord 𝑁 → Tr 𝑁) | |
2 | ordelord 6391 | . . . . 5 ⊢ ((Ord 𝑁 ∧ 𝑥 ∈ 𝑁) → Ord 𝑥) | |
3 | ordtr 6383 | . . . . 5 ⊢ (Ord 𝑥 → Tr 𝑥) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((Ord 𝑁 ∧ 𝑥 ∈ 𝑁) → Tr 𝑥) |
5 | 4 | ralrimiva 3143 | . . 3 ⊢ (Ord 𝑁 → ∀𝑥 ∈ 𝑁 Tr 𝑥) |
6 | 1, 5 | jca 511 | . 2 ⊢ (Ord 𝑁 → (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥)) |
7 | simpl 482 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → Tr 𝑁) | |
8 | dford3lem1 42447 | . . . . 5 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → ∀𝑎 ∈ 𝑁 (Tr 𝑎 ∧ ∀𝑥 ∈ 𝑎 Tr 𝑥)) | |
9 | dford3lem2 42448 | . . . . . 6 ⊢ ((Tr 𝑎 ∧ ∀𝑥 ∈ 𝑎 Tr 𝑥) → 𝑎 ∈ On) | |
10 | 9 | ralimi 3080 | . . . . 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 3968 | . . . 4 ⊢ (𝑁 ⊆ On ↔ ∀𝑎 ∈ 𝑁 𝑎 ∈ On) | |
13 | 11, 12 | sylibr 233 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → 𝑁 ⊆ On) |
14 | ordon 7779 | . . . 4 ⊢ Ord On | |
15 | 14 | a1i 11 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → Ord On) |
16 | trssord 6386 | . . 3 ⊢ ((Tr 𝑁 ∧ 𝑁 ⊆ On ∧ Ord On) → Ord 𝑁) | |
17 | 7, 13, 15, 16 | syl3anc 1369 | . 2 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → Ord 𝑁) |
18 | 6, 17 | impbii 208 | 1 ⊢ (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2099 ∀wral 3058 ⊆ wss 3947 Tr wtr 5265 Ord word 6368 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-pr 5429 ax-un 7740 ax-reg 9616 |
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-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-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-br 5149 df-opab 5211 df-tr 5266 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6372 df-on 6373 df-suc 6375 |
This theorem is referenced by: dford4 42450 |
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