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Theorem dford5 33137
 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 7497 . . 3 (Ord 𝐴𝐴 ⊆ On)
2 ordtr 6180 . . 3 (Ord 𝐴 → Tr 𝐴)
31, 2jca 515 . 2 (Ord 𝐴 → (𝐴 ⊆ On ∧ Tr 𝐴))
4 epweon 7490 . . . 4 E We On
5 wess 5510 . . . 4 (𝐴 ⊆ On → ( E We On → E We 𝐴))
64, 5mpi 20 . . 3 (𝐴 ⊆ On → E We 𝐴)
7 df-ord 6169 . . . . 5 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
87biimpri 231 . . . 4 ((Tr 𝐴 ∧ E We 𝐴) → Ord 𝐴)
98ancoms 462 . . 3 (( E We 𝐴 ∧ Tr 𝐴) → Ord 𝐴)
106, 9sylan 583 . 2 ((𝐴 ⊆ On ∧ Tr 𝐴) → Ord 𝐴)
113, 10impbii 212 1 (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ⊆ wss 3883  Tr wtr 5140   E cep 5433   We wwe 5481  Ord word 6165  Oncon0 6166 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-tr 5141  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6169  df-on 6170 This theorem is referenced by:  nosupno  33393  noinfno  33408
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