MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dftr5 Structured version   Visualization version   GIF version

Theorem dftr5 4720
Description: An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
Assertion
Ref Expression
dftr5 (Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dftr5
StepHypRef Expression
1 dftr2 4719 . 2 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2 alcom 2039 . . 3 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
3 impexp 462 . . . . . . . 8 (((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ (𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
43albii 1744 . . . . . . 7 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
5 df-ral 2917 . . . . . . 7 (∀𝑦𝑥 (𝑥𝐴𝑦𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
64, 5bitr4i 267 . . . . . 6 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑦𝑥 (𝑥𝐴𝑦𝐴))
7 r19.21v 2959 . . . . . 6 (∀𝑦𝑥 (𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
86, 7bitri 264 . . . . 5 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
98albii 1744 . . . 4 (∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
10 df-ral 2917 . . . 4 (∀𝑥𝐴𝑦𝑥 𝑦𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
119, 10bitr4i 267 . . 3 (∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
122, 11bitri 264 . 2 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
131, 12bitri 264 1 (Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wcel 1992  wral 2912  Tr wtr 4717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-v 3193  df-in 3567  df-ss 3574  df-uni 4408  df-tr 4718
This theorem is referenced by:  dftr3  4721  smobeth  9353
  Copyright terms: Public domain W3C validator