MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ontrci Structured version   Visualization version   GIF version

Theorem ontrci 5976
Description: An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
ontrci Tr 𝐴

Proof of Theorem ontrci
StepHypRef Expression
1 on.1 . . 3 𝐴 ∈ On
21onordi 5975 . 2 Ord 𝐴
3 ordtr 5880 . 2 (Ord 𝐴 → Tr 𝐴)
42, 3ax-mp 5 1 Tr 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2144  Tr wtr 4884  Ord word 5865  Oncon0 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-v 3351  df-in 3728  df-ss 3735  df-uni 4573  df-tr 4885  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870
This theorem is referenced by:  onunisuci  5984  hfuni  32622
  Copyright terms: Public domain W3C validator