Theorem dftr2 2678

Description: An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40.

Assertion

Ref	Expression
dftr2	|- (Tr A <-> A.xA.y((x e. y /\ y e. A) -> x e. A))

Distinct variable group:   x,y,A

Proof of Theorem dftr2

Step	Hyp	Ref	Expression
1		dfss2 2055	. 2 |- (U.A (_ A <-> A.x(x e. U.A -> x e. A))
2		df-tr 2677	. 2 |- (Tr A <-> U.A (_ A)
3		19.23v 1292	. . . 4 |- (A.y((x e. y /\ y e. A) -> x e. A) <-> (E.y(x e. y /\ y e. A) -> x e. A))
4		eluni 2502	. . . . 5 |- (x e. U.A <-> E.y(x e. y /\ y e. A))
5	4	imbi1i 186	. . . 4 |- ((x e. U.A -> x e. A) <-> (E.y(x e. y /\ y e. A) -> x e. A))
6	3, 5	bitr4 176	. . 3 |- (A.y((x e. y /\ y e. A) -> x e. A) <-> (x e. U.A -> x e. A))
7	6	albii 998	. 2 |- (A.xA.y((x e. y /\ y e. A) -> x e. A) <-> A.x(x e. U.A -> x e. A))
8	1, 2, 7	3bitr4 183	1 |- (Tr A <-> A.xA.y((x e. y /\ y e. A) -> x e. A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   e. wcel 957  E.wex 979   (_ wss 2044  U.cuni 2499  Tr wtr 2676

This theorem is referenced by:  dftr5 2679  trel 2683  ordelord 2966  ordom 3137  trcl 4628  ondomon 4839

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-v 1809  df-in 2048  df-ss 2050  df-uni 2500  df-tr 2677

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