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Theorem ontrci 5515
Description: An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
Hypothesis
Ref Expression
on.1  |-  A  e.  On
Assertion
Ref Expression
ontrci  |-  Tr  A

Proof of Theorem ontrci
StepHypRef Expression
1 on.1 . . 3  |-  A  e.  On
21onordi 5514 . 2  |-  Ord  A
3 ordtr 5424 . 2  |-  ( Ord 
A  ->  Tr  A
)
42, 3ax-mp 5 1  |-  Tr  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1842   Tr wtr 4489   Ord word 5409   Oncon0 5410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-v 3061  df-in 3421  df-ss 3428  df-uni 4192  df-tr 4490  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 5413  df-on 5414
This theorem is referenced by:  onunisuci  5523  hfuni  30522
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