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Theorem dftr6 31345
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 5326 . . . 4 (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴)
3 brdif 4665 . . . . . 6 (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ (𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴))
4 vex 3189 . . . . . . . . 9 𝑥 ∈ V
54, 1brco 5252 . . . . . . . 8 (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥 E 𝑦𝑦 E 𝐴))
6 epel 4988 . . . . . . . . . 10 (𝑥 E 𝑦𝑥𝑦)
71epelc 4987 . . . . . . . . . 10 (𝑦 E 𝐴𝑦𝐴)
86, 7anbi12i 732 . . . . . . . . 9 ((𝑥 E 𝑦𝑦 E 𝐴) ↔ (𝑥𝑦𝑦𝐴))
98exbii 1771 . . . . . . . 8 (∃𝑦(𝑥 E 𝑦𝑦 E 𝐴) ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
105, 9bitri 264 . . . . . . 7 (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
111epelc 4987 . . . . . . . 8 (𝑥 E 𝐴𝑥𝐴)
1211notbii 310 . . . . . . 7 𝑥 E 𝐴 ↔ ¬ 𝑥𝐴)
1310, 12anbi12i 732 . . . . . 6 ((𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴))
14 19.41v 1911 . . . . . . 7 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴))
15 exanali 1783 . . . . . . 7 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴) ↔ ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
1614, 15bitr3i 266 . . . . . 6 ((∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴) ↔ ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
173, 13, 163bitri 286 . . . . 5 (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
1817exbii 1771 . . . 4 (∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ∃𝑥 ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
19 exnal 1751 . . . 4 (∃𝑥 ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ¬ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
202, 18, 193bitri 286 . . 3 (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ¬ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
2120con2bii 347 . 2 (∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
22 dftr2 4714 . 2 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
23 eldif 3565 . . 3 (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ (𝐴 ∈ V ∧ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E )))
241, 23mpbiran 952 . 2 (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
2521, 22, 243bitr4i 292 1 (Tr 𝐴𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478  wex 1701  wcel 1987  Vcvv 3186  cdif 3552   class class class wbr 4613  Tr wtr 4712   E cep 4983  ran crn 5075  ccom 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085
This theorem is referenced by:  eltrans  31637
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