Theorem treq 2682

Description: Equality theorem for the transitive class predicate.

Assertion

Ref	Expression
treq	|- (A = B -> (Tr A <-> Tr B))

Proof of Theorem treq

Step	Hyp	Ref	Expression
1		unieq 2506	. . . 4 |- (A = B -> U.A = U.B)
2	1	sseq1d 2085	. . 3 |- (A = B -> (U.A (_ A <-> U.B (_ A))
3		sseq2 2080	. . 3 |- (A = B -> (U.B (_ A <-> U.B (_ B))
4	2, 3	bitrd 527	. 2 |- (A = B -> (U.A (_ A <-> U.B (_ B))
5		df-tr 2677	. 2 |- (Tr A <-> U.A (_ A)
6		df-tr 2677	. 2 |- (Tr B <-> U.B (_ B)
7	4, 5, 6	3bitr4g 554	1 |- (A = B -> (Tr A <-> Tr B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955   (_ wss 2044  U.cuni 2499  Tr wtr 2676

This theorem is referenced by:  ordeq 2951  trcl 4628  tz9.1 4629  r1tr 4637

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-in 2048  df-ss 2050  df-uni 2500  df-tr 2677

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