Theorem dftr4 2046

Description: An alternate way of defining a transitive class. Definition of [Enderton] p. 71.

Assertion

Ref	Expression
dftr4	⊢ (Tr A ↔ A ⊆ ℘A)

Proof of Theorem dftr4

Step	Hyp	Ref	Expression
1		visset 1350	. . . 4 ⊢ x ∈ V
2	1	elpw 1801	. . 3 ⊢ (x ∈ ℘A ↔ x ⊆ A)
3	2	biral 1223	. 2 ⊢ (∀x ∈ A x ∈ ℘A ↔ ∀x ∈ A x ⊆ A)
4		dfss3 1498	. 2 ⊢ (A ⊆ ℘A ↔ ∀x ∈ A x ∈ ℘A)
5		dftr3 2045	. 2 ⊢ (Tr A ↔ ∀x ∈ A x ⊆ A)
6	3, 4, 5	3bitr4r 159	1 ⊢ (Tr A ↔ A ⊆ ℘A)

Syntax hints:   ↔ wb 127   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  ℘cpw 1798  Tr wtr 2041

This theorem is referenced by:  tr0 2052  r1tr 3498

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-9799  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-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-pw 1799  df-uni 1920  df-tr 2042

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