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Theorem dftr3 4684
 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 4683 . 2 (Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
2 dfss3 3558 . . 3 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
32ralbii 2963 . 2 (∀𝑥𝐴 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
41, 3bitr4i 266 1 (Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  Tr wtr 4680 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373  df-tr 4681 This theorem is referenced by:  trss  4689  trssOLD  4690  trin  4691  triun  4694  trint  4696  tron  5663  ssorduni  6877  suceloni  6905  dfrecs3  7356  ordtypelem2  8307  tcwf  8629  itunitc  9126  wunex2  9439  wfgru  9517
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