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Theorem ordon 4370
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 4272 . 2  |-  Tr  On
2 df-on 4258 . . . . 5  |-  On  =  { x  |  Ord  x }
32abeq2i 2226 . . . 4  |-  ( x  e.  On  <->  Ord  x )
4 ordtr 4268 . . . 4  |-  ( Ord  x  ->  Tr  x
)
53, 4sylbi 120 . . 3  |-  ( x  e.  On  ->  Tr  x )
65rgen 2460 . 2  |-  A. x  e.  On  Tr  x
7 dford3 4257 . 2  |-  ( Ord 
On 
<->  ( Tr  On  /\  A. x  e.  On  Tr  x ) )
81, 6, 7mpbir2an 909 1  |-  Ord  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1463   A.wral 2391   Tr wtr 3994   Ord word 4252   Oncon0 4253
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-in 3045  df-ss 3052  df-uni 3705  df-tr 3995  df-iord 4256  df-on 4258
This theorem is referenced by:  ssorduni  4371  limon  4397  onprc  4435  tfri1dALT  6214  rdgon  6249
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