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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 3966 . . 3 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
32ralbii 3088 . 2 (∀𝑥𝐴 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
41, 3bitr4i 278 1 (Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2099  wral 3056  wss 3944  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-v 3471  df-in 3951  df-ss 3961  df-uni 4904  df-tr 5260
This theorem is referenced by:  trss  5270  trin  5271  triun  5274  triin  5276  tron  6386  ssorduni  7775  sucexeloniOLD  7807  suceloniOLD  7809  dfrecs3  8386  dfrecs3OLD  8387  ordtypelem2  9536  tcwf  9900  itunitc  10438  wunex2  10755  wfgru  10833  nadd2rabtr  42785
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