Theorem treq 4546

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 4253	. . . 4 |- ( A = B -> U. A = U. B )
2	1	sseq1d 3531	. . 3 |- ( A = B -> ( U. A C_ A <-> U. B C_ A ) )
3		sseq2 3526	. . 3 |- ( A = B -> ( U. B C_ A <-> U. B C_ B ) )
4	2, 3	bitrd 253	. 2 |- ( A = B -> ( U. A C_ A <-> U. B C_ B ) )
5		df-tr 4541	. 2 |- ( Tr A <-> U. A C_ A )
6		df-tr 4541	. 2 |- ( Tr B <-> U. B C_ B )
7	4, 5, 6	3bitr4g 288	1 |- ( A = B -> ( Tr A <-> Tr B ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1379   C_ wss 3476   U.cuni 4245   Tr wtr 4540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-in 3483  df-ss 3490  df-uni 4246  df-tr 4541

This theorem is referenced by:  truni  4554  ordeq  4885  trcl  8159  tz9.1  8160  tz9.1c  8161  tctr  8171  tcmin  8172  tc2  8173  r1tr  8194  r1elssi  8223  tcrank  8302  iswun  9082  tskr1om2  9146  elgrug  9170  grutsk  9200  dfon2lem1  28820  dfon2lem3  28822  dfon2lem4  28823  dfon2lem5  28824  dfon2lem6  28825  dfon2lem7  28826  dfon2lem8  28827  dfon2  28829  dford3lem1  30600  dford3lem2  30601