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Theorem eltrans 35520
Description: Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
Hypothesis
Ref Expression
eltrans.1 𝐴 ∈ V
Assertion
Ref Expression
eltrans (𝐴 Trans ↔ Tr 𝐴)

Proof of Theorem eltrans
StepHypRef Expression
1 df-trans 35486 . . 3 Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
21eleq2i 2821 . 2 (𝐴 Trans 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
3 eltrans.1 . . 3 𝐴 ∈ V
43dftr6 35378 . 2 (Tr 𝐴𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
52, 4bitr4i 277 1 (𝐴 Trans ↔ Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098  Vcvv 3473  cdif 3946  Tr wtr 5269   E cep 5585  ran crn 5683  ccom 5686   Trans ctrans 35462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-trans 35486
This theorem is referenced by:  dfon3  35521
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