Theorem treq 4389

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 4097	. . . 4 |- ( A = B -> U. A = U. B )
2	1	sseq1d 3381	. . 3 |- ( A = B -> ( U. A C_ A <-> U. B C_ A ) )
3		sseq2 3376	. . 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 4384	. 2 |- ( Tr A <-> U. A C_ A )
6		df-tr 4384	. 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 1369   C_ wss 3326   U.cuni 4089   Tr wtr 4383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-in 3333  df-ss 3340  df-uni 4090  df-tr 4384

This theorem is referenced by:  truni  4397  ordeq  4724  trcl  7946  tz9.1  7947  tz9.1c  7948  tctr  7958  tcmin  7959  tc2  7960  r1tr  7981  r1elssi  8010  tcrank  8089  iswun  8869  tskr1om2  8933  elgrug  8957  grutsk  8987  dfon2lem1  27594  dfon2lem3  27596  dfon2lem4  27597  dfon2lem5  27598  dfon2lem6  27599  dfon2lem7  27600  dfon2lem8  27601  dfon2  27603  dford3lem1  29372  dford3lem2  29373