Theorem treq 4164

Description: Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)

Assertion

Ref	Expression
treq	|- ( A = B -> ( Tr A <-> Tr B ) )

Proof of Theorem treq

Step	Hyp	Ref	Expression
1		unieq 3873	. . . 4 |- ( A = B -> U. A = U. B )
2	1	sseq1d 3230	. . 3 |- ( A = B -> ( U. A C_ A <-> U. B C_ A ) )
3		sseq2 3225	. . 3 |- ( A = B -> ( U. B C_ A <-> U. B C_ B ) )
4	2, 3	bitrd 188	. 2 |- ( A = B -> ( U. A C_ A <-> U. B C_ B ) )
5		df-tr 4159	. 2 |- ( Tr A <-> U. A C_ A )
6		df-tr 4159	. 2 |- ( Tr B <-> U. B C_ B )
7	4, 5, 6	3bitr4g 223	1 |- ( A = B -> ( Tr A <-> Tr B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   C_ wss 3174   U.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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  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-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159

This theorem is referenced by:  truni  4172  ordeq  4437  ordsucim  4566  ordom  4673  exmidonfinlem  7332