Theorem treq 4268

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 3984	. . . 4 |- ( A = B -> U. A = U. B )
2	1	sseq1d 3335	. . 3 |- ( A = B -> ( U. A C_ A <-> U. B C_ A ) )
3		sseq2 3330	. . 3 |- ( A = B -> ( U. B C_ A <-> U. B C_ B ) )
4	2, 3	bitrd 245	. 2 |- ( A = B -> ( U. A C_ A <-> U. B C_ B ) )
5		df-tr 4263	. 2 |- ( Tr A <-> U. A C_ A )
6		df-tr 4263	. 2 |- ( Tr B <-> U. B C_ B )
7	4, 5, 6	3bitr4g 280	1 |- ( A = B -> ( Tr A <-> Tr B ) )

Syntax hints:   -> wi 4   <-> wb 177   = wceq 1649   C_ wss 3280   U.cuni 3975   Tr wtr 4262

This theorem is referenced by:  truni  4276  ordeq  4548  trcl  7620  tz9.1  7621  tz9.1c  7622  tctr  7635  tcmin  7636  tc2  7637  r1tr  7658  r1elssi  7687  tcrank  7764  iswun  8535  tskr1om2  8599  elgrug  8623  grutsk  8653  dfon2lem1  25353  dfon2lem3  25355  dfon2lem4  25356  dfon2lem5  25357  dfon2lem6  25358  dfon2lem7  25359  dfon2lem8  25360  dfon2  25362  dford3lem1  26987  dford3lem2  26988

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-in 3287  df-ss 3294  df-uni 3976  df-tr 4263