Theorem trv 4153

Description: The universe is transitive. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
trv	⊢ Tr V

Proof of Theorem trv

Step	Hyp	Ref	Expression
1		ssv 3214	. 2 ⊢ ∪ V ⊆ V
2		df-tr 4142	. 2 ⊢ (Tr V ↔ ∪ V ⊆ V)
3	1, 2	mpbir 146	1 ⊢ Tr V

Syntax hints:  Vcvv 2771   ⊆ wss 3165  ∪ cuni 3849  Tr wtr 4141

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-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-v 2773  df-in 3171  df-ss 3178  df-tr 4142

This theorem is referenced by: (None)