Theorem ordtr 4081

Description: An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)

Assertion

Ref	Expression
ordtr	⊢ (Ord A → Tr A)

Proof of Theorem ordtr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dford3 4070	. 2 ⊢ (Ord A ↔ (Tr A ∧ ∀x ∈ A Tr x))
2	1	simplbi 259	1 ⊢ (Ord A → Tr A)

Syntax hints:   → wi 4  ∀wral 2300  Tr wtr 3845  Ord word 4065

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99

This theorem depends on definitions:  df-bi 110  df-iord 4069

This theorem is referenced by:  ordelss  4082  ordin  4088  ordtr1  4091  orduniss  4128  ontrci  4130  ordon  4178  ordsucim  4192  ordsucss  4196  onsucsssucr  4200  ordsucunielexmid  4216  nlimsucg  4242  ordpwsucss  4243  tfrexlem  5889