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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 5654 | . . 3 ⊢ (Ord 𝑁 → Tr 𝑁) | |
2 | ordelord 5662 | . . . . 5 ⊢ ((Ord 𝑁 ∧ 𝑥 ∈ 𝑁) → Ord 𝑥) | |
3 | ordtr 5654 | . . . . 5 ⊢ (Ord 𝑥 → Tr 𝑥) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((Ord 𝑁 ∧ 𝑥 ∈ 𝑁) → Tr 𝑥) |
5 | 4 | ralrimiva 2949 | . . 3 ⊢ (Ord 𝑁 → ∀𝑥 ∈ 𝑁 Tr 𝑥) |
6 | 1, 5 | jca 553 | . 2 ⊢ (Ord 𝑁 → (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥)) |
7 | simpl 472 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → Tr 𝑁) | |
8 | dford3lem1 36611 | . . . . 5 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → ∀𝑎 ∈ 𝑁 (Tr 𝑎 ∧ ∀𝑥 ∈ 𝑎 Tr 𝑥)) | |
9 | dford3lem2 36612 | . . . . . 6 ⊢ ((Tr 𝑎 ∧ ∀𝑥 ∈ 𝑎 Tr 𝑥) → 𝑎 ∈ On) | |
10 | 9 | ralimi 2936 | . . . . 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 3558 | . . . 4 ⊢ (𝑁 ⊆ On ↔ ∀𝑎 ∈ 𝑁 𝑎 ∈ On) | |
13 | 11, 12 | sylibr 223 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → 𝑁 ⊆ On) |
14 | ordon 6874 | . . . 4 ⊢ Ord On | |
15 | 14 | a1i 11 | . . 3 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → Ord On) |
16 | trssord 5657 | . . 3 ⊢ ((Tr 𝑁 ∧ 𝑁 ⊆ On ∧ Ord On) → Ord 𝑁) | |
17 | 7, 13, 15, 16 | syl3anc 1318 | . 2 ⊢ ((Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥) → Ord 𝑁) |
18 | 6, 17 | impbii 198 | 1 ⊢ (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 Tr wtr 4680 Ord word 5639 Oncon0 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 ax-reg 8380 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-suc 5646 |
This theorem is referenced by: dford4 36614 |
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