Theorem dftr4 5217

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 5212	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
2		sspwuni 5059	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
3	1, 2	bitr4i 281	1 ⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ⊆ wss 3895  𝒫 cpw 4556  ∪ cuni 4866  Tr wtr 5211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3896  df-pw 4558  df-uni 4867  df-tr 5212

This theorem is used by:  tr0  5224  pwtr  5419  r1ordg  9788  r1sssuc  9793  r1val1  9796  ackbij2lem3  10321  tsktrss  10849  par2catid  50860