| 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 6398 | . . 3 ⊢ (Ord 𝑁 → Tr 𝑁) | |
| 2 | ordelord 6406 | . . . . 5 ⊢ ((Ord 𝑁 ∧ 𝑥 ∈ 𝑁) → Ord 𝑥) | |
| 3 | ordtr 6398 | . . . . 5 ⊢ (Ord 𝑥 → Tr 𝑥) | |
| 4 | 2, 3 | syl 17 | . . . 4 ⊢ ((Ord 𝑁 ∧ 𝑥 ∈ 𝑁) → Tr 𝑥) |
| 5 | 4 | ralrimiva 3146 | . . 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 43038 | . . . . 5 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → ∀𝑎 ∈ 𝑁 (Tr 𝑎 ∧ ∀𝑥 ∈ 𝑎 Tr 𝑥)) | |
| 9 | dford3lem2 43039 | . . . . . 6 ⊢ ((Tr 𝑎 ∧ ∀𝑥 ∈ 𝑎 Tr 𝑥) → 𝑎 ∈ On) | |
| 10 | 9 | ralimi 3083 | . . . . 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 3972 | . . . 4 ⊢ (𝑁 ⊆ On ↔ ∀𝑎 ∈ 𝑁 𝑎 ∈ On) | |
| 13 | 11, 12 | sylibr 234 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → 𝑁 ⊆ On) |
| 14 | ordon 7797 | . . . 4 ⊢ Ord On | |
| 15 | 14 | a1i 11 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → Ord On) |
| 16 | trssord 6401 | . . 3 ⊢ ((Tr 𝑁 ∧ 𝑁 ⊆ On ∧ Ord On) → Ord 𝑁) | |
| 17 | 7, 13, 15, 16 | syl3anc 1373 | . 2 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → Ord 𝑁) |
| 18 | 6, 17 | impbii 209 | 1 ⊢ (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 Tr wtr 5259 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-reg 9632 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 |
| This theorem is referenced by: dford4 43041 |
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