Theorem eltrans 35855

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

Step	Hyp	Ref	Expression
1		df-trans 35821	. . 3 ⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
2	1	eleq2i 2826	. 2 ⊢ (𝐴 ∈ Trans ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
3		eltrans.1	. . 3 ⊢ 𝐴 ∈ V
4	3	dftr6 35714	. 2 ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
5	2, 4	bitr4i 278	1 ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  Vcvv 3459   ∖ cdif 3923  Tr wtr 5229   E cep 5552  ran crn 5655   ∘ ccom 5658   Trans ctrans 35797

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-tr 5230  df-eprel 5553  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-trans 35821

This theorem is referenced by:  dfon3  35856