Theorem dftr2 2043

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(s):   x,y,A

Proof of Theorem dftr2

Step	Hyp	Ref	Expression
1		dfss2 1497	. 2 |- (U.A (_ A <-> A.x(x e. U.A -> x e. A))
2		df-tr 2042	. 2 |- (Tr A <-> U.A (_ A)
3		19.23v 950	. . . 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 1922	. . . . 5 |- (x e. U.A <-> E.y(x e. y /\ y e. A))
5	4	imbi1i 161	. . . 4 |- ((x e. U.A -> x e. A) <-> (E.y(x e. y /\ y e. A) -> x e. A))
6	3, 5	bitr4 154	. . 3 |- (A.y((x e. y /\ y e. A) -> x e. A) <-> (x e. U.A -> x e. A))
7	6	bial 695	. 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 158	1 |- (Tr A <-> A.xA.y((x e. y /\ y e. A) -> x e. A))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   e. wel 803   e. wcel 1092   (_ wss 1487  U.cuni 1919  Tr wtr 2041

This theorem is referenced by:  dftr5 2044  trel 2048  ordelord 2221  ordom 2382  tfrlem8 2956  trcl 3489  ondomon 3662

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920  df-tr 2042

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