Theorem dftr6 33718

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 5802	. . . 4 ⊢ (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴)
3		brdif 5127	. . . . . 6 ⊢ (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ (𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴))
4		vex 3436	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5	4, 1	brco 5779	. . . . . . . 8 ⊢ (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥 E 𝑦 ∧ 𝑦 E 𝐴))
6		epel 5498	. . . . . . . . . 10 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
7	1	epeli 5497	. . . . . . . . . 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 5497	. . . . . . . 8 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)
12	11	notbii 320	. . . . . . 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 297	. . . . 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 297	. . 3 ⊢ (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ¬ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
21	20	con2bii 358	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
22		dftr2 5193	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
23		eldif 3897	. . 3 ⊢ (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ (𝐴 ∈ V ∧ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E )))
24	1, 23	mpbiran 706	. 2 ⊢ (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
25	21, 22, 24	3bitr4i 303	1 ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537  ∃wex 1782   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884   class class class wbr 5074  Tr wtr 5191   E cep 5494  ran crn 5590   ∘ ccom 5593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600

This theorem is referenced by:  eltrans  34193