Theorem dftr6 34324

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 5849	. . . 4 ⊢ (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴)
3		brdif 5158	. . . . . 6 ⊢ (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ (𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴))
4		vex 3449	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5	4, 1	brco 5826	. . . . . . . 8 ⊢ (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥 E 𝑦 ∧ 𝑦 E 𝐴))
6		epel 5540	. . . . . . . . . 10 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
7	1	epeli 5539	. . . . . . . . . 10 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)
8	6, 7	anbi12i 627	. . . . . . . . 9 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝐴) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
9	8	exbii 1850	. . . . . . . 8 ⊢ (∃𝑦(𝑥 E 𝑦 ∧ 𝑦 E 𝐴) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
10	5, 9	bitri 274	. . . . . . 7 ⊢ (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
11	1	epeli 5539	. . . . . . . 8 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)
12	11	notbii 319	. . . . . . 7 ⊢ (¬ 𝑥 E 𝐴 ↔ ¬ 𝑥 ∈ 𝐴)
13	10, 12	anbi12i 627	. . . . . 6 ⊢ ((𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑥 ∈ 𝐴))
14		19.41v 1953	. . . . . . 7 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑥 ∈ 𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑥 ∈ 𝐴))
15		exanali 1862	. . . . . . 7 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑥 ∈ 𝐴) ↔ ¬ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
16	14, 15	bitr3i 276	. . . . . 6 ⊢ ((∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑥 ∈ 𝐴) ↔ ¬ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
17	3, 13, 16	3bitri 296	. . . . 5 ⊢ (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ¬ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
18	17	exbii 1850	. . . 4 ⊢ (∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ∃𝑥 ¬ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
19		exnal 1829	. . . 4 ⊢ (∃𝑥 ¬ ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ¬ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
20	2, 18, 19	3bitri 296	. . 3 ⊢ (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ¬ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
21	20	con2bii 357	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
22		dftr2 5224	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
23		eldif 3920	. . 3 ⊢ (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ (𝐴 ∈ V ∧ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E )))
24	1, 23	mpbiran 707	. 2 ⊢ (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
25	21, 22, 24	3bitr4i 302	1 ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539  ∃wex 1781   ∈ wcel 2106  Vcvv 3445   ∖ cdif 3907   class class class wbr 5105  Tr wtr 5222   E cep 5536  ran crn 5634   ∘ ccom 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-tr 5223  df-eprel 5537  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644

This theorem is referenced by:  eltrans  34476