Theorem treq 3860

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 3589	. . . 4 |- ( A = B -> U. A = U. B )
2	1	sseq1d 2972	. . 3 |- ( A = B -> ( U. A C_ A <-> U. B C_ A ) )
3		sseq2 2967	. . 3 |- ( A = B -> ( U. B C_ A <-> U. B C_ B ) )
4	2, 3	bitrd 177	. 2 |- ( A = B -> ( U. A C_ A <-> U. B C_ B ) )
5		df-tr 3855	. 2 |- ( Tr A <-> U. A C_ A )
6		df-tr 3855	. 2 |- ( Tr B <-> U. B C_ B )
7	4, 5, 6	3bitr4g 212	1 |- ( A = B -> ( Tr A <-> Tr B ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   C_ wss 2917   U.cuni 3580   Tr wtr 3854

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855

This theorem is referenced by:  truni  3868  ordeq  4109  ordsucim  4226  ordom  4329