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Theorem dftr6 31918
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 5509 . . . 4 (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴)
3 brdif 4845 . . . . . 6 (𝑥(( E ∘ E ) ∖ E )𝐴 ↔ (𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴))
4 vex 3331 . . . . . . . . 9 𝑥 ∈ V
54, 1brco 5436 . . . . . . . 8 (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥 E 𝑦𝑦 E 𝐴))
6 epel 5170 . . . . . . . . . 10 (𝑥 E 𝑦𝑥𝑦)
71epelc 5169 . . . . . . . . . 10 (𝑦 E 𝐴𝑦𝐴)
86, 7anbi12i 735 . . . . . . . . 9 ((𝑥 E 𝑦𝑦 E 𝐴) ↔ (𝑥𝑦𝑦𝐴))
98exbii 1911 . . . . . . . 8 (∃𝑦(𝑥 E 𝑦𝑦 E 𝐴) ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
105, 9bitri 264 . . . . . . 7 (𝑥( E ∘ E )𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
111epelc 5169 . . . . . . . 8 (𝑥 E 𝐴𝑥𝐴)
1211notbii 309 . . . . . . 7 𝑥 E 𝐴 ↔ ¬ 𝑥𝐴)
1310, 12anbi12i 735 . . . . . 6 ((𝑥( E ∘ E )𝐴 ∧ ¬ 𝑥 E 𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴))
14 19.41v 2014 . . . . . . 7 (∃𝑦((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ 𝑥𝐴))
15 exanali 1923 . . . . . . 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 1911 . . . 4 (∃𝑥 𝑥(( E ∘ E ) ∖ E )𝐴 ↔ ∃𝑥 ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
19 exnal 1891 . . . 4 (∃𝑥 ¬ ∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ¬ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
202, 18, 193bitri 286 . . 3 (𝐴 ∈ ran (( E ∘ E ) ∖ E ) ↔ ¬ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
2120con2bii 346 . 2 (∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E ))
22 dftr2 4894 . 2 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
23 eldif 3713 . . 3 (𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )) ↔ (𝐴 ∈ V ∧ ¬ 𝐴 ∈ ran (( E ∘ E ) ∖ E )))
241, 23mpbiran 991 . 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 383  wal 1618  wex 1841  wcel 2127  Vcvv 3328  cdif 3700   class class class wbr 4792  Tr wtr 4892   E cep 5166  ran crn 5255  ccom 5258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-tr 4893  df-eprel 5167  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265
This theorem is referenced by:  eltrans  32275
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