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Theorem ordon 6979
Description: The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
Assertion
Ref Expression
ordon Ord On

Proof of Theorem ordon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tron 5744 . 2 Tr On
2 onfr 5761 . . 3 E Fr On
3 eloni 5731 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
4 eloni 5731 . . . . 5 (𝑦 ∈ On → Ord 𝑦)
5 ordtri3or 5753 . . . . . 6 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
6 epel 5030 . . . . . . 7 (𝑥 E 𝑦𝑥𝑦)
7 biid 251 . . . . . . 7 (𝑥 = 𝑦𝑥 = 𝑦)
8 epel 5030 . . . . . . 7 (𝑦 E 𝑥𝑦𝑥)
96, 7, 83orbi123i 1251 . . . . . 6 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
105, 9sylibr 224 . . . . 5 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
113, 4, 10syl2an 494 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
1211rgen2a 2976 . . 3 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)
13 dfwe2 6978 . . 3 ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
142, 12, 13mpbir2an 955 . 2 E We On
15 df-ord 5724 . 2 (Ord On ↔ (Tr On ∧ E We On))
161, 14, 15mpbir2an 955 1 Ord On
Colors of variables: wff setvar class
Syntax hints:  wa 384  w3o 1036  wcel 1989  wral 2911   class class class wbr 4651  Tr wtr 4750   E cep 5026   Fr wfr 5068   We wwe 5070  Ord word 5720  Oncon0 5721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-tr 4751  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-ord 5724  df-on 5725
This theorem is referenced by:  epweon  6980  onprc  6981  ssorduni  6982  ordeleqon  6985  ordsson  6986  onint  6992  suceloni  7010  limon  7033  tfi  7050  ordom  7071  ordtypelem2  8421  hartogs  8446  card2on  8456  tskwe  8773  alephsmo  8922  ondomon  9382  dford3lem2  37420  dford3  37421  iunord  42193
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