Theorem trv 4092

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 3164	. 2 ⊢ ∪ V ⊆ V
2		df-tr 4081	. 2 ⊢ (Tr V ↔ ∪ V ⊆ V)
3	1, 2	mpbir 145	1 ⊢ Tr V

Syntax hints:  Vcvv 2726   ⊆ wss 3116  ∪ cuni 3789  Tr wtr 4080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728  df-in 3122  df-ss 3129  df-tr 4081

This theorem is referenced by: (None)