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

Theorem sspsstri 3693
Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
sspsstri ((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Proof of Theorem sspsstri
StepHypRef Expression
1 or32 549 . 2 (((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
2 sspss 3690 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
3 sspss 3690 . . . . 5 (𝐵𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
4 eqcom 2628 . . . . . 6 (𝐵 = 𝐴𝐴 = 𝐵)
54orbi2i 541 . . . . 5 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐵𝐴𝐴 = 𝐵))
63, 5bitri 264 . . . 4 (𝐵𝐴 ↔ (𝐵𝐴𝐴 = 𝐵))
72, 6orbi12i 543 . . 3 ((𝐴𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ (𝐵𝐴𝐴 = 𝐵)))
8 orordir 553 . . 3 (((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ (𝐵𝐴𝐴 = 𝐵)))
97, 8bitr4i 267 . 2 ((𝐴𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵))
10 df-3or 1037 . 2 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
111, 9, 103bitr4i 292 1 ((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  w3o 1035   = wceq 1480  wss 3560  wpss 3561
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ne 2791  df-in 3567  df-ss 3574  df-pss 3576
This theorem is referenced by:  ordtri3or  5724  sorpss  6907  sorpssi  6908  funpsstri  31420
  Copyright terms: Public domain W3C validator