Theorem dftr4 5211

Description: An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)

Assertion

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

Proof of Theorem dftr4

Step	Hyp	Ref	Expression
1		df-tr 5206	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
2		sspwuni 5055	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
3	1, 2	bitr4i 278	1 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)

Syntax hints:   ↔ wb 206   ⊆ wss 3901  𝒫 cpw 4554  ∪ cuni 4863  Tr wtr 5205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3442  df-ss 3918  df-pw 4556  df-uni 4864  df-tr 5206

This theorem is referenced by:  tr0  5217  pwtr  5400  r1ordg  9690  r1sssuc  9695  r1val1  9698  ackbij2lem3  10150  tsktrss  10672