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Theorem dford5 31836
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 7106 . . 3 (Ord 𝐴𝐴 ⊆ On)
2 ordtr 5850 . . 3 (Ord 𝐴 → Tr 𝐴)
31, 2jca 555 . 2 (Ord 𝐴 → (𝐴 ⊆ On ∧ Tr 𝐴))
4 epweon 7100 . . . 4 E We On
5 wess 5205 . . . 4 (𝐴 ⊆ On → ( E We On → E We 𝐴))
64, 5mpi 20 . . 3 (𝐴 ⊆ On → E We 𝐴)
7 df-ord 5839 . . . . 5 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
87biimpri 218 . . . 4 ((Tr 𝐴 ∧ E We 𝐴) → Ord 𝐴)
98ancoms 468 . . 3 (( E We 𝐴 ∧ Tr 𝐴) → Ord 𝐴)
106, 9sylan 489 . 2 ((𝐴 ⊆ On ∧ Tr 𝐴) → Ord 𝐴)
113, 10impbii 199 1 (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wss 3680  Tr wtr 4860   E cep 5132   We wwe 5176  Ord word 5835  Oncon0 5836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-tr 4861  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-ord 5839  df-on 5840
This theorem is referenced by:  nosupno  32076
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