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Theorem ordon 4447
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 4344 . 2  |-  Tr  On
2 df-on 4330 . . . . 5  |-  On  =  { x  |  Ord  x }
32abeq2i 2268 . . . 4  |-  ( x  e.  On  <->  Ord  x )
4 ordtr 4340 . . . 4  |-  ( Ord  x  ->  Tr  x
)
53, 4sylbi 120 . . 3  |-  ( x  e.  On  ->  Tr  x )
65rgen 2510 . 2  |-  A. x  e.  On  Tr  x
7 dford3 4329 . 2  |-  ( Ord 
On 
<->  ( Tr  On  /\  A. x  e.  On  Tr  x ) )
81, 6, 7mpbir2an 927 1  |-  Ord  On
Colors of variables: wff set class
Syntax hints:    e. wcel 2128   A.wral 2435   Tr wtr 4064   Ord word 4324   Oncon0 4325
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-uni 3775  df-tr 4065  df-iord 4328  df-on 4330
This theorem is referenced by:  ssorduni  4448  limon  4474  onprc  4513  tfri1dALT  6300  rdgon  6335
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