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Theorem dftr6 34709
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 5891 . . . 4 (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴)
3 brdif 5200 . . . . . 6 (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ (𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴))
4 vex 3478 . . . . . . . . 9 𝑥 ∈ V
54, 1brco 5868 . . . . . . . 8 (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥 E 𝑦𝑦 E 𝐴))
6 epel 5582 . . . . . . . . . 10 (𝑥 E 𝑦𝑥𝑦)
71epeli 5581 . . . . . . . . . 10 (𝑦 E 𝐴𝑦𝐴)
86, 7anbi12i 627 . . . . . . . . 9 ((𝑥 E 𝑦𝑦 E 𝐴) ↔ (𝑥𝑦𝑦𝐴))
98exbii 1850 . . . . . . . 8 (∃𝑦(𝑥 E 𝑦𝑦 E 𝐴) ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
105, 9bitri 274 . . . . . . 7 (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
111epeli 5581 . . . . . . . 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 1953 . . . . . . 7 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴))
15 exanali 1862 . . . . . . 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 1850 . . . 4 (∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ∃𝑥 ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
19 exnal 1829 . . . 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 5266 . 2 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
23 eldif 3957 . . 3 (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ (𝐴 ∈ V ∧ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E )))
241, 23mpbiran 707 . 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 1539  wex 1781  wcel 2106  Vcvv 3474  cdif 3944   class class class wbr 5147  Tr wtr 5264   E cep 5578  ran crn 5676  ccom 5679
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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
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 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686
This theorem is referenced by:  eltrans  34851
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