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Theorem eltrans 34407
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 34373 . . 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 34118 . 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 205  wcel 2107  Vcvv 3444  cdif 3906  Tr wtr 5221   E cep 5534  ran crn 5632  ccom 5635   Trans ctrans 34349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-tr 5222  df-eprel 5535  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-trans 34373
This theorem is referenced by:  dfon3  34408
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