Theorem eltrans 36476

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 36442	. . 3 ⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
2	1	eleq2i 2854	. 2 ⊢ (𝐴 ∈ Trans ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
3		eltrans.1	. . 3 ⊢ 𝐴 ∈ V
4	3	dftr6 36338	. 2 ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
5	2, 4	bitr4i 281	1 ⊢ (𝐴 ∈ Trans ↔ Tr 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2145  Vcvv 3453   ∖ cdif 3899  Tr wtr 5216   E cep 5558  ran crn 5660   ∘ ccom 5663   Trans ctrans 36418

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  df-eprel 5559  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-trans 36442

This theorem is used by:  dfon3  36477