Theorem eltrans 35924

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 35890	. . 3 ⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
2	1	eleq2i 2823	. 2 ⊢ (𝐴 ∈ Trans ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
3		eltrans.1	. . 3 ⊢ 𝐴 ∈ V
4	3	dftr6 35783	. 2 ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
5	2, 4	bitr4i 278	1 ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3899  Tr wtr 5198   E cep 5515  ran crn 5617   ∘ ccom 5620   Trans ctrans 35866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-eprel 5516  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-trans 35890

This theorem is referenced by:  dfon3  35925