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Theorem dftr3 5265
Description: An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)
Assertion
Ref Expression
dftr3 (Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem dftr3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dftr5 5263 . 2 (Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
2 dfss3 3972 . . 3 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
32ralbii 3093 . 2 (∀𝑥𝐴 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
41, 3bitr4i 278 1 (Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2108  wral 3061  wss 3951  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260
This theorem is referenced by:  trss  5270  trin  5271  triun  5274  triin  5276  tron  6407  ssorduni  7799  sucexeloniOLD  7830  suceloniOLD  7832  dfrecs3  8412  dfrecs3OLD  8413  ordtypelem2  9559  tcwf  9923  itunitc  10461  wunex2  10778  wfgru  10856  nadd2rabtr  43397  trwf  44976
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