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Theorem dftr6 34007
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 𝐴𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))

Proof of Theorem dftr6
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr6.1 . . . . 5 𝐴 ∈ V
21elrn 5835 . . . 4 (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴)
3 brdif 5145 . . . . . 6 (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ (𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴))
4 vex 3445 . . . . . . . . 9 𝑥 ∈ V
54, 1brco 5812 . . . . . . . 8 (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥 E 𝑦𝑦 E 𝐴))
6 epel 5527 . . . . . . . . . 10 (𝑥 E 𝑦𝑥𝑦)
71epeli 5526 . . . . . . . . . 10 (𝑦 E 𝐴𝑦𝐴)
86, 7anbi12i 627 . . . . . . . . 9 ((𝑥 E 𝑦𝑦 E 𝐴) ↔ (𝑥𝑦𝑦𝐴))
98exbii 1849 . . . . . . . 8 (∃𝑦(𝑥 E 𝑦𝑦 E 𝐴) ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
105, 9bitri 274 . . . . . . 7 (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
111epeli 5526 . . . . . . . 8 (𝑥 E 𝐴𝑥𝐴)
1211notbii 319 . . . . . . 7 𝑥 E 𝐴 ↔ ¬ 𝑥𝐴)
1310, 12anbi12i 627 . . . . . 6 ((𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴))
14 19.41v 1952 . . . . . . 7 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴))
15 exanali 1861 . . . . . . 7 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴) ↔ ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
1614, 15bitr3i 276 . . . . . 6 ((∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴) ↔ ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
173, 13, 163bitri 296 . . . . 5 (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
1817exbii 1849 . . . 4 (∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ∃𝑥 ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
19 exnal 1828 . . . 4 (∃𝑥 ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ¬ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
202, 18, 193bitri 296 . . 3 (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ¬ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
2120con2bii 357 . 2 (∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
22 dftr2 5211 . 2 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
23 eldif 3908 . . 3 (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ (𝐴 ∈ V ∧ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E )))
241, 23mpbiran 706 . 2 (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
2521, 22, 243bitr4i 302 1 (Tr 𝐴𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1538  wex 1780  wcel 2105  Vcvv 3441  cdif 3895   class class class wbr 5092  Tr wtr 5209   E cep 5523  ran crn 5621  ccom 5624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-tr 5210  df-eprel 5524  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631
This theorem is referenced by:  eltrans  34289
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