ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dftr2 GIF version

Theorem dftr2 3960
Description: An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)
Assertion
Ref Expression
dftr2 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dftr2
StepHypRef Expression
1 dfss2 3028 . 2 ( 𝐴𝐴 ↔ ∀𝑥(𝑥 𝐴𝑥𝐴))
2 df-tr 3959 . 2 (Tr 𝐴 𝐴𝐴)
3 19.23v 1818 . . . 4 (∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) → 𝑥𝐴))
4 eluni 3678 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
54imbi1i 237 . . . 4 ((𝑥 𝐴𝑥𝐴) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) → 𝑥𝐴))
63, 5bitr4i 186 . . 3 (∀𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ (𝑥 𝐴𝑥𝐴))
76albii 1411 . 2 (∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴) ↔ ∀𝑥(𝑥 𝐴𝑥𝐴))
81, 2, 73bitr4i 211 1 (Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1294  wex 1433  wcel 1445  wss 3013   cuni 3675  Tr wtr 3958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-uni 3676  df-tr 3959
This theorem is referenced by:  dftr5  3961  trel  3965  suctr  4272  ordtriexmidlem  4364  ordtri2or2exmidlem  4370  onsucelsucexmidlem  4373  ordsuc  4407  tfi  4425  ordom  4449
  Copyright terms: Public domain W3C validator