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

Theorem dftr5 4990
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 4989 . 2 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2 alcom 2151 . . 3 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
3 impexp 443 . . . . . . . 8 (((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ (𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
43albii 1863 . . . . . . 7 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
5 df-ral 3094 . . . . . . 7 (∀𝑦𝑥 (𝑥𝐴𝑦𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
64, 5bitr4i 270 . . . . . 6 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑦𝑥 (𝑥𝐴𝑦𝐴))
7 r19.21v 3141 . . . . . 6 (∀𝑦𝑥 (𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
86, 7bitri 267 . . . . 5 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
98albii 1863 . . . 4 (∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
10 df-ral 3094 . . . 4 (∀𝑥𝐴𝑦𝑥 𝑦𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
119, 10bitr4i 270 . . 3 (∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
122, 11bitri 267 . 2 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
131, 12bitri 267 1 (Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wal 1599  wcel 2106  wral 3089  Tr wtr 4987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-v 3399  df-in 3798  df-ss 3805  df-uni 4672  df-tr 4988
This theorem is referenced by:  dftr3  4991  smobeth  9743
  Copyright terms: Public domain W3C validator