Theorem dftr6

Description: A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012)

	Ref	Expression
Hypothesis	dftr6.1	⊢ 𝐴 ∈ V
Assertion	dftr6	⊢ ( Tr 𝐴 ↔ 𝐴 ∈ ( V ∖ ran ( ( E ∘ E ) ∖ E ) ) )

Proof

Step	Hyp	Ref	Expression
1		dftr6.1	⊢ 𝐴 ∈ V
2	1	elrn	⊢ ( 𝐴 ∈ ran ( ( E ∘ E ) ∖ E ) ↔ ∃ 𝑥 𝑥 ( ( E ∘ E ) ∖ E ) 𝐴 )
3		brdif	⊢ ( 𝑥 ( ( E ∘ E ) ∖ E ) 𝐴 ↔ ( 𝑥 ( E ∘ E ) 𝐴 ∧ ¬ 𝑥 E 𝐴 ) )
4		vex	⊢ 𝑥 ∈ V
5	4 1	brco	⊢ ( 𝑥 ( E ∘ E ) 𝐴 ↔ ∃ 𝑦 ( 𝑥 E 𝑦 ∧ 𝑦 E 𝐴 ) )
6		epel	⊢ ( 𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦 )
7	1	epeli	⊢ ( 𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴 )
8	6 7	anbi12i	⊢ ( ( 𝑥 E 𝑦 ∧ 𝑦 E 𝐴 ) ↔ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) )
9	8	exbii	⊢ ( ∃ 𝑦 ( 𝑥 E 𝑦 ∧ 𝑦 E 𝐴 ) ↔ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) )
10	5 9	bitri	⊢ ( 𝑥 ( E ∘ E ) 𝐴 ↔ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) )
11	1	epeli	⊢ ( 𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴 )
12	11	notbii	⊢ ( ¬ 𝑥 E 𝐴 ↔ ¬ 𝑥 ∈ 𝐴 )
13	10 12	anbi12i	⊢ ( ( 𝑥 ( E ∘ E ) 𝐴 ∧ ¬ 𝑥 E 𝐴 ) ↔ ( ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑥 ∈ 𝐴 ) )
14		19.41v	⊢ ( ∃ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑥 ∈ 𝐴 ) ↔ ( ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑥 ∈ 𝐴 ) )
15		exanali	⊢ ( ∃ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑥 ∈ 𝐴 ) ↔ ¬ ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) )
16	14 15	bitr3i	⊢ ( ( ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∧ ¬ 𝑥 ∈ 𝐴 ) ↔ ¬ ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) )
17	3 13 16	3bitri	⊢ ( 𝑥 ( ( E ∘ E ) ∖ E ) 𝐴 ↔ ¬ ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) )
18	17	exbii	⊢ ( ∃ 𝑥 𝑥 ( ( E ∘ E ) ∖ E ) 𝐴 ↔ ∃ 𝑥 ¬ ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) )
19		exnal	⊢ ( ∃ 𝑥 ¬ ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) ↔ ¬ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) )
20	2 18 19	3bitri	⊢ ( 𝐴 ∈ ran ( ( E ∘ E ) ∖ E ) ↔ ¬ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) )
21	20	con2bii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) ↔ ¬ 𝐴 ∈ ran ( ( E ∘ E ) ∖ E ) )
22		dftr2	⊢ ( Tr 𝐴 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) )
23		eldif	⊢ ( 𝐴 ∈ ( V ∖ ran ( ( E ∘ E ) ∖ E ) ) ↔ ( 𝐴 ∈ V ∧ ¬ 𝐴 ∈ ran ( ( E ∘ E ) ∖ E ) ) )
24	1 23	mpbiran	⊢ ( 𝐴 ∈ ( V ∖ ran ( ( E ∘ E ) ∖ E ) ) ↔ ¬ 𝐴 ∈ ran ( ( E ∘ E ) ∖ E ) )
25	21 22 24	3bitr4i	⊢ ( Tr 𝐴 ↔ 𝐴 ∈ ( V ∖ ran ( ( E ∘ E ) ∖ E ) ) )