Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  distel Structured version   Visualization version   GIF version

Theorem distel 32006
Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4988 and elirrv 8658.) (Contributed by Scott Fenton, 15-Dec-2010.)
Assertion
Ref Expression
distel (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦)

Proof of Theorem distel
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 el 4988 . . 3 𝑧 𝑥𝑧
2 df-ex 1846 . . . 4 (∃𝑧 𝑥𝑧 ↔ ¬ ∀𝑧 ¬ 𝑥𝑧)
3 nfnae 2452 . . . . . 6 𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4 dveel1 2499 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥𝑧 → ∀𝑦 𝑥𝑧))
53, 4nf5d 2257 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥𝑧)
65nfnd 1926 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 ¬ 𝑥𝑧)
7 elequ2 2145 . . . . . . . 8 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
87notbid 307 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑥𝑧 ↔ ¬ 𝑥𝑦))
98a1i 11 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → (¬ 𝑥𝑧 ↔ ¬ 𝑥𝑦)))
103, 6, 9cbvald 2414 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑧 ¬ 𝑥𝑧 ↔ ∀𝑦 ¬ 𝑥𝑦))
1110notbid 307 . . . 4 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑧 ¬ 𝑥𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦))
122, 11syl5bb 272 . . 3 (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑧 𝑥𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦))
131, 12mpbii 223 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 ¬ 𝑥𝑦)
14 elirrv 8658 . . . . 5 ¬ 𝑦𝑦
15 elequ1 2138 . . . . 5 (𝑦 = 𝑥 → (𝑦𝑦𝑥𝑦))
1614, 15mtbii 315 . . . 4 (𝑦 = 𝑥 → ¬ 𝑥𝑦)
1716alimi 1880 . . 3 (∀𝑦 𝑦 = 𝑥 → ∀𝑦 ¬ 𝑥𝑦)
1817con3i 150 . 2 (¬ ∀𝑦 ¬ 𝑥𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
1913, 18impbii 199 1 (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1622  wex 1845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-reg 8654
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-v 3334  df-dif 3710  df-un 3712  df-nul 4051  df-sn 4314  df-pr 4316
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator