Theorem trv 4170

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 3223	. 2 ⊢ ∪ V ⊆ V
2		df-tr 4159	. 2 ⊢ (Tr V ↔ ∪ V ⊆ V)
3	1, 2	mpbir 146	1 ⊢ Tr V

Syntax hints:  Vcvv 2776   ⊆ wss 3174  ∪ cuni 3864  Tr wtr 4158

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778  df-in 3180  df-ss 3187  df-tr 4159

This theorem is referenced by: (None)