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Theorem eltrans 35395
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 35361 . . 3 Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
21eleq2i 2819 . 2 (𝐴 Trans 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
3 eltrans.1 . . 3 𝐴 ∈ V
43dftr6 35253 . 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 2098  Vcvv 3468  cdif 3940  Tr wtr 5258   E cep 5572  ran crn 5670  ccom 5673   Trans ctrans 35337
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-trans 35361
This theorem is referenced by:  dfon3  35396
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