Theorem treq 4198

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 3907	. . . 4 |- ( A = B -> U. A = U. B )
2	1	sseq1d 3257	. . 3 |- ( A = B -> ( U. A C_ A <-> U. B C_ A ) )
3		sseq2 3252	. . 3 |- ( A = B -> ( U. B C_ A <-> U. B C_ B ) )
4	2, 3	bitrd 188	. 2 |- ( A = B -> ( U. A C_ A <-> U. B C_ B ) )
5		df-tr 4193	. 2 |- ( Tr A <-> U. A C_ A )
6		df-tr 4193	. 2 |- ( Tr B <-> U. B C_ B )
7	4, 5, 6	3bitr4g 223	1 |- ( A = B -> ( Tr A <-> Tr B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   C_ wss 3201   U.cuni 3898   Tr wtr 4192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-in 3207  df-ss 3214  df-uni 3899  df-tr 4193

This theorem is referenced by:  truni  4206  ordeq  4475  ordsucim  4604  ordom  4711  exmidonfinlem  7464