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Theorem ordon 4470
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
StepHypRef Expression
1 tron 4367 . 2  |-  Tr  On
2 df-on 4353 . . . . 5  |-  On  =  { x  |  Ord  x }
32abeq2i 2281 . . . 4  |-  ( x  e.  On  <->  Ord  x )
4 ordtr 4363 . . . 4  |-  ( Ord  x  ->  Tr  x
)
53, 4sylbi 120 . . 3  |-  ( x  e.  On  ->  Tr  x )
65rgen 2523 . 2  |-  A. x  e.  On  Tr  x
7 dford3 4352 . 2  |-  ( Ord 
On 
<->  ( Tr  On  /\  A. x  e.  On  Tr  x ) )
81, 6, 7mpbir2an 937 1  |-  Ord  On
Colors of variables: wff set class
Syntax hints:    e. wcel 2141   A.wral 2448   Tr wtr 4087   Ord word 4347   Oncon0 4348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353
This theorem is referenced by:  ssorduni  4471  limon  4497  onprc  4536  tfri1dALT  6330  rdgon  6365
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