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Theorem dftr3 4915
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 4914 . 2 (Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
2 dfss3 3750 . . 3 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
32ralbii 3127 . 2 (∀𝑥𝐴 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
41, 3bitr4i 269 1 (Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wcel 2155  wral 3055  wss 3732  Tr wtr 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-v 3352  df-in 3739  df-ss 3746  df-uni 4595  df-tr 4912
This theorem is referenced by:  trss  4920  trin  4921  triun  4924  triin  4926  trintOLD  4928  tron  5931  ssorduni  7183  suceloni  7211  dfrecs3  7673  ordtypelem2  8631  tcwf  8961  itunitc  9496  wunex2  9813  wfgru  9891
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