Theorem dftr2 2682

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 2058	. 2 |- (U.A (_ A <-> A.x(x e. U.A -> x e. A))
2		df-tr 2681	. 2 |- (Tr A <-> U.A (_ A)
3		19.23v 1293	. . . 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 2506	. . . . 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 999	. 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 954   e. wcel 958  E.wex 980   (_ wss 2047  U.cuni 2503  Tr wtr 2680

This theorem is referenced by:  dftr5 2683  trel 2687  ordelord 2970  ordom 3141  trcl 4645  ondomon 4856

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-uni 2504  df-tr 2681

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