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Theorem eltrans 35872
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 35838 . . 3 Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
21eleq2i 2830 . 2 (𝐴 Trans 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
3 eltrans.1 . . 3 𝐴 ∈ V
43dftr6 35730 . 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 2105  Vcvv 3477  cdif 3959  Tr wtr 5264   E cep 5587  ran crn 5689  ccom 5692   Trans ctrans 35814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-trans 35838
This theorem is referenced by:  dfon3  35873
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