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Theorem eltrans 35892
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 35858 . . 3 Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
21eleq2i 2833 . 2 (𝐴 Trans 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
3 eltrans.1 . . 3 𝐴 ∈ V
43dftr6 35751 . 2 (Tr 𝐴𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
52, 4bitr4i 278 1 (𝐴 Trans ↔ Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2108  Vcvv 3480  cdif 3948  Tr wtr 5259   E cep 5583  ran crn 5686  ccom 5689   Trans ctrans 35834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-trans 35858
This theorem is referenced by:  dfon3  35893
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