Theorem dftr4 5221

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 5216	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
2		sspwuni 5063	. 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 3900  𝒫 cpw 4560  ∪ cuni 4870  Tr wtr 5215

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-ss 3901  df-pw 4562  df-uni 4871  df-tr 5216

This theorem is used by:  tr0  5228  pwtr  5424  r1ordg  9812  r1sssuc  9817  r1val1  9820  ackbij2lem3  10345  tsktrss  10873  par2catid  50972