Theorem ontrci 4456

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

Step	Hyp	Ref	Expression
1		on.1	. . 3 |- A e. On
2	1	onordi 4455	. 2 |- Ord A
3		ordtr 4407	. 2 |- ( Ord A -> Tr A )
4	2, 3	ax-mp 5	1 |- Tr A

Syntax hints:   e. wcel 2164   Tr wtr 4127   Ord word 4391   Oncon0 4392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-uni 3836  df-tr 4128  df-iord 4395  df-on 4397

This theorem is referenced by:  onunisuci  4461  exmidonfinlem  7247  bj-el2oss1o  15244  nnsf  15473