Theorem treq 4040

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 3753	. . . 4 |- ( A = B -> U. A = U. B )
2	1	sseq1d 3131	. . 3 |- ( A = B -> ( U. A C_ A <-> U. B C_ A ) )
3		sseq2 3126	. . 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 4035	. 2 |- ( Tr A <-> U. A C_ A )
6		df-tr 4035	. 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 1332   C_ wss 3076   U.cuni 3744   Tr wtr 4034

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-in 3082  df-ss 3089  df-uni 3745  df-tr 4035

This theorem is referenced by:  truni  4048  ordeq  4302  ordsucim  4424  ordom  4528  exmidonfinlem  7066