Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dftr4 Structured version   Visualization version   GIF version

Theorem dftr4 5144
 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 𝐴𝐴 ⊆ 𝒫 𝐴)

Proof of Theorem dftr4
StepHypRef Expression
1 df-tr 5140 . 2 (Tr 𝐴 𝐴𝐴)
2 sspwuni 4988 . 2 (𝐴 ⊆ 𝒫 𝐴 𝐴𝐴)
31, 2bitr4i 281 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ⊆ wss 3859  𝒫 cpw 4495  ∪ cuni 4799  Tr wtr 5139 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2159  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-v 3412  df-in 3866  df-ss 3876  df-pw 4497  df-uni 4800  df-tr 5140 This theorem is referenced by:  tr0  5150  pwtr  5314  r1ordg  9241  r1sssuc  9246  r1val1  9249  ackbij2lem3  9702  tsktrss  10222
 Copyright terms: Public domain W3C validator