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Theorem dford3 37072
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 5696 . . 3 (Ord 𝑁 → Tr 𝑁)
2 ordelord 5704 . . . . 5 ((Ord 𝑁𝑥𝑁) → Ord 𝑥)
3 ordtr 5696 . . . . 5 (Ord 𝑥 → Tr 𝑥)
42, 3syl 17 . . . 4 ((Ord 𝑁𝑥𝑁) → Tr 𝑥)
54ralrimiva 2960 . . 3 (Ord 𝑁 → ∀𝑥𝑁 Tr 𝑥)
61, 5jca 554 . 2 (Ord 𝑁 → (Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥))
7 simpl 473 . . 3 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → Tr 𝑁)
8 dford3lem1 37070 . . . . 5 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → ∀𝑎𝑁 (Tr 𝑎 ∧ ∀𝑥𝑎 Tr 𝑥))
9 dford3lem2 37071 . . . . . 6 ((Tr 𝑎 ∧ ∀𝑥𝑎 Tr 𝑥) → 𝑎 ∈ On)
109ralimi 2947 . . . . 5 (∀𝑎𝑁 (Tr 𝑎 ∧ ∀𝑥𝑎 Tr 𝑥) → ∀𝑎𝑁 𝑎 ∈ On)
118, 10syl 17 . . . 4 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → ∀𝑎𝑁 𝑎 ∈ On)
12 dfss3 3573 . . . 4 (𝑁 ⊆ On ↔ ∀𝑎𝑁 𝑎 ∈ On)
1311, 12sylibr 224 . . 3 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → 𝑁 ⊆ On)
14 ordon 6929 . . . 4 Ord On
1514a1i 11 . . 3 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → Ord On)
16 trssord 5699 . . 3 ((Tr 𝑁𝑁 ⊆ On ∧ Ord On) → Ord 𝑁)
177, 13, 15, 16syl3anc 1323 . 2 ((Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥) → Ord 𝑁)
186, 17impbii 199 1 (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wcel 1987  wral 2907  wss 3555  Tr wtr 4712  Ord word 5681  Oncon0 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902  ax-reg 8441
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686  df-suc 5688
This theorem is referenced by:  dford4  37073
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