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Theorem dftr3 4716
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 4715 . 2 (Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
2 dfss3 3573 . . 3 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
32ralbii 2974 . 2 (∀𝑥𝐴 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
41, 3bitr4i 267 1 (Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1987  wral 2907  wss 3555  Tr wtr 4712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-in 3562  df-ss 3569  df-uni 4403  df-tr 4713
This theorem is referenced by:  trss  4721  trssOLD  4722  trin  4723  triun  4726  trint  4728  tron  5705  ssorduni  6932  suceloni  6960  dfrecs3  7414  ordtypelem2  8368  tcwf  8690  itunitc  9187  wunex2  9504  wfgru  9582
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