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Theorem dford5 7805
Description: A class is ordinal iff it is a subclass of On and transitive. (Contributed by Scott Fenton, 21-Nov-2021.)
Assertion
Ref Expression
dford5 (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴))

Proof of Theorem dford5
StepHypRef Expression
1 ordsson 7804 . . 3 (Ord 𝐴𝐴 ⊆ On)
2 ordtr 6397 . . 3 (Ord 𝐴 → Tr 𝐴)
31, 2jca 511 . 2 (Ord 𝐴 → (𝐴 ⊆ On ∧ Tr 𝐴))
4 epweon 7796 . . . 4 E We On
5 wess 5670 . . . 4 (𝐴 ⊆ On → ( E We On → E We 𝐴))
64, 5mpi 20 . . 3 (𝐴 ⊆ On → E We 𝐴)
7 df-ord 6386 . . . . 5 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
87biimpri 228 . . . 4 ((Tr 𝐴 ∧ E We 𝐴) → Ord 𝐴)
98ancoms 458 . . 3 (( E We 𝐴 ∧ Tr 𝐴) → Ord 𝐴)
106, 9sylan 580 . 2 ((𝐴 ⊆ On ∧ Tr 𝐴) → Ord 𝐴)
113, 10impbii 209 1 (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wss 3950  Tr wtr 5258   E cep 5582   We wwe 5635  Ord word 6382  Oncon0 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387
This theorem is referenced by:  nosupno  27749  noinfno  27764  nadd2rabord  43403  nadd1rabord  43407
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