Theorem dftr6 30893

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

Hypothesis

Ref	Expression
dftr6.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
dftr6	⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))

Proof of Theorem dftr6

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr6.1	. . . . 5 ⊢ 𝐴 ∈ V
2	1	elrn 5287	. . . 4 ⊢ (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴)
3		brdif 4635	. . . . . 6 ⊢ (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ (𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴))
4		vex 3176	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5	4, 1	brco 5214	. . . . . . . 8 ⊢ (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥 E 𝑦 ∧ 𝑦 E 𝐴))
6		epel 4952	. . . . . . . . . 10 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
7	1	epelc 4951	. . . . . . . . . 10 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)
8	6, 7	anbi12i 729	. . . . . . . . 9 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝐴) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
9	8	exbii 1764	. . . . . . . 8 ⊢ (∃𝑦(𝑥 E 𝑦 ∧ 𝑦 E 𝐴) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
10	5, 9	bitri 263	. . . . . . 7 ⊢ (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
11	1	epelc 4951	. . . . . . . 8 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)
12	11	notbii 309	. . . . . . 7 ⊢ (¬ 𝑥 E 𝐴 ↔ ¬ 𝑥 ∈ 𝐴)
13	10, 12	anbi12i 729	. . . . . 6 ⊢ ((𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑥 ∈ 𝐴))
14		19.41v 1901	. . . . . . 7 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑥 ∈ 𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑥 ∈ 𝐴))
15		exanali 1773	. . . . . . 7 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑥 ∈ 𝐴) ↔ ¬ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
16	14, 15	bitr3i 265	. . . . . 6 ⊢ ((∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑥 ∈ 𝐴) ↔ ¬ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
17	3, 13, 16	3bitri 285	. . . . 5 ⊢ (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ¬ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
18	17	exbii 1764	. . . 4 ⊢ (∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ∃𝑥 ¬ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
19		exnal 1744	. . . 4 ⊢ (∃𝑥 ¬ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ¬ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
20	2, 18, 19	3bitri 285	. . 3 ⊢ (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ¬ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
21	20	con2bii 346	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
22		dftr2 4682	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
23		eldif 3550	. . 3 ⊢ (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ (𝐴 ∈ V ∧ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E )))
24	1, 23	mpbiran 955	. 2 ⊢ (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
25	21, 22, 24	3bitr4i 291	1 ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   class class class wbr 4583  Tr wtr 4680   E cep 4947  ran crn 5039   ∘ ccom 5042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049

This theorem is referenced by:  eltrans  31168