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Theorem dford5 7770
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 7769 . . 3 (Ord 𝐴𝐴 ⊆ On)
2 ordtr 6378 . . 3 (Ord 𝐴 → Tr 𝐴)
31, 2jca 512 . 2 (Ord 𝐴 → (𝐴 ⊆ On ∧ Tr 𝐴))
4 epweon 7761 . . . 4 E We On
5 wess 5663 . . . 4 (𝐴 ⊆ On → ( E We On → E We 𝐴))
64, 5mpi 20 . . 3 (𝐴 ⊆ On → E We 𝐴)
7 df-ord 6367 . . . . 5 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
87biimpri 227 . . . 4 ((Tr 𝐴 ∧ E We 𝐴) → Ord 𝐴)
98ancoms 459 . . 3 (( E We 𝐴 ∧ Tr 𝐴) → Ord 𝐴)
106, 9sylan 580 . 2 ((𝐴 ⊆ On ∧ Tr 𝐴) → Ord 𝐴)
113, 10impbii 208 1 (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wss 3948  Tr wtr 5265   E cep 5579   We wwe 5630  Ord word 6363  Oncon0 6364
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368
This theorem is referenced by:  nosupno  27203  noinfno  27218  nadd2rabord  42125  nadd1rabord  42129
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