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Theorem dftr6 32088
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 5537 . . . 4 (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴)
3 brdif 4864 . . . . . 6 (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ (𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴))
4 vex 3353 . . . . . . . . 9 𝑥 ∈ V
54, 1brco 5463 . . . . . . . 8 (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥 E 𝑦𝑦 E 𝐴))
6 epel 5195 . . . . . . . . . 10 (𝑥 E 𝑦𝑥𝑦)
71epeli 5194 . . . . . . . . . 10 (𝑦 E 𝐴𝑦𝐴)
86, 7anbi12i 620 . . . . . . . . 9 ((𝑥 E 𝑦𝑦 E 𝐴) ↔ (𝑥𝑦𝑦𝐴))
98exbii 1943 . . . . . . . 8 (∃𝑦(𝑥 E 𝑦𝑦 E 𝐴) ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
105, 9bitri 266 . . . . . . 7 (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
111epeli 5194 . . . . . . . 8 (𝑥 E 𝐴𝑥𝐴)
1211notbii 311 . . . . . . 7 𝑥 E 𝐴 ↔ ¬ 𝑥𝐴)
1310, 12anbi12i 620 . . . . . 6 ((𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴))
14 19.41v 2044 . . . . . . 7 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴))
15 exanali 1955 . . . . . . 7 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴) ↔ ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
1614, 15bitr3i 268 . . . . . 6 ((∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴) ↔ ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
173, 13, 163bitri 288 . . . . 5 (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
1817exbii 1943 . . . 4 (∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ∃𝑥 ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
19 exnal 1921 . . . 4 (∃𝑥 ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ¬ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
202, 18, 193bitri 288 . . 3 (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ¬ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
2120con2bii 348 . 2 (∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
22 dftr2 4915 . 2 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
23 eldif 3744 . . 3 (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ (𝐴 ∈ V ∧ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E )))
241, 23mpbiran 700 . 2 (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
2521, 22, 243bitr4i 294 1 (Tr 𝐴𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wal 1650  wex 1874  wcel 2155  Vcvv 3350  cdif 3731   class class class wbr 4811  Tr wtr 4913   E cep 5191  ran crn 5280  ccom 5283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-tr 4914  df-eprel 5192  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290
This theorem is referenced by:  eltrans  32445
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