Theorem treq 3881

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 3610	. . . 4 |- ( A = B -> U. A = U. B )
2	1	sseq1d 3026	. . 3 |- ( A = B -> ( U. A C_ A <-> U. B C_ A ) )
3		sseq2 3021	. . 3 |- ( A = B -> ( U. B C_ A <-> U. B C_ B ) )
4	2, 3	bitrd 186	. 2 |- ( A = B -> ( U. A C_ A <-> U. B C_ B ) )
5		df-tr 3876	. 2 |- ( Tr A <-> U. A C_ A )
6		df-tr 3876	. 2 |- ( Tr B <-> U. B C_ B )
7	4, 5, 6	3bitr4g 221	1 |- ( A = B -> ( Tr A <-> Tr B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   C_ wss 2973   U.cuni 3601   Tr wtr 3875

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876

This theorem is referenced by:  truni  3889  ordeq  4127  ordsucim  4244  ordom  4347