Theorem treq 3940

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 3660	. . . 4 |- ( A = B -> U. A = U. B )
2	1	sseq1d 3053	. . 3 |- ( A = B -> ( U. A C_ A <-> U. B C_ A ) )
3		sseq2 3048	. . 3 |- ( A = B -> ( U. B C_ A <-> U. B C_ B ) )
4	2, 3	bitrd 186	. 2 |- ( A = B -> ( U. A C_ A <-> U. B C_ B ) )
5		df-tr 3935	. 2 |- ( Tr A <-> U. A C_ A )
6		df-tr 3935	. 2 |- ( Tr B <-> U. B C_ B )
7	4, 5, 6	3bitr4g 221	1 |- ( A = B -> ( Tr A <-> Tr B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   C_ wss 2999   U.cuni 3651   Tr wtr 3934

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-in 3005  df-ss 3012  df-uni 3652  df-tr 3935

This theorem is referenced by:  truni  3948  ordeq  4197  ordsucim  4315  ordom  4419