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Theorem dford3 36613
 Description: Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)
Assertion
Ref Expression
dford3 (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥))
Distinct variable group:   𝑥,𝑁

Proof of Theorem dford3
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 ordtr 5654 . . 3 (Ord 𝑁 → Tr 𝑁)
2 ordelord 5662 . . . . 5 ((Ord 𝑁𝑥𝑁) → Ord 𝑥)
3 ordtr 5654 . . . . 5 (Ord 𝑥 → Tr 𝑥)
42, 3syl 17 . . . 4 ((Ord 𝑁𝑥𝑁) → Tr 𝑥)
54ralrimiva 2949 . . 3 (Ord 𝑁 → ∀𝑥𝑁 Tr 𝑥)
61, 5jca 553 . 2 (Ord 𝑁 → (Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥))
7 simpl 472 . . 3 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → Tr 𝑁)
8 dford3lem1 36611 . . . . 5 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → ∀𝑎𝑁 (Tr 𝑎 ∧ ∀𝑥𝑎 Tr 𝑥))
9 dford3lem2 36612 . . . . . 6 ((Tr 𝑎 ∧ ∀𝑥𝑎 Tr 𝑥) → 𝑎 ∈ On)
109ralimi 2936 . . . . 5 (∀𝑎𝑁 (Tr 𝑎 ∧ ∀𝑥𝑎 Tr 𝑥) → ∀𝑎𝑁 𝑎 ∈ On)
118, 10syl 17 . . . 4 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → ∀𝑎𝑁 𝑎 ∈ On)
12 dfss3 3558 . . . 4 (𝑁 ⊆ On ↔ ∀𝑎𝑁 𝑎 ∈ On)
1311, 12sylibr 223 . . 3 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → 𝑁 ⊆ On)
14 ordon 6874 . . . 4 Ord On
1514a1i 11 . . 3 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → Ord On)
16 trssord 5657 . . 3 ((Tr 𝑁𝑁 ⊆ On ∧ Ord On) → Ord 𝑁)
177, 13, 15, 16syl3anc 1318 . 2 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → Ord 𝑁)
186, 17impbii 198 1 (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥𝑁 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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