Theorem treq 4032

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 3745	. . . 4 |- ( A = B -> U. A = U. B )
2	1	sseq1d 3126	. . 3 |- ( A = B -> ( U. A C_ A <-> U. B C_ A ) )
3		sseq2 3121	. . 3 |- ( A = B -> ( U. B C_ A <-> U. B C_ B ) )
4	2, 3	bitrd 187	. 2 |- ( A = B -> ( U. A C_ A <-> U. B C_ B ) )
5		df-tr 4027	. 2 |- ( Tr A <-> U. A C_ A )
6		df-tr 4027	. 2 |- ( Tr B <-> U. B C_ B )
7	4, 5, 6	3bitr4g 222	1 |- ( A = B -> ( Tr A <-> Tr B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   C_ wss 3071   U.cuni 3736   Tr wtr 4026

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027

This theorem is referenced by:  truni  4040  ordeq  4294  ordsucim  4416  ordom  4520  exmidonfinlem  7049