Theorem ordtr 4064

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 4053	. 2 ⊢ (Ord A ↔ (Tr A ∧ ∀x ∈ A Tr x))
2	1	simplbi 259	1 ⊢ (Ord A → Tr A)

Syntax hints:   → wi 4  ∀wral 2284  Tr wtr 3828  Ord word 4048

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 4052

This theorem is referenced by:  ordelss  4065  ordin  4071  ordtr1  4074  orduniss  4112  ontrci  4114  ordon  4162  ordsucim  4176  ordsucss  4180  onsucsssucr  4184  ordsucunielexmid  4200  nlimsucg  4226  ordpwsucss  4227  tfrexlem  5870