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

Theorem sspsstrir 3023
 Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by Jim Kingdon, 16-Jul-2018.)
Assertion
Ref Expression
sspsstrir ((AB A = B BA) → (AB BA))

Proof of Theorem sspsstrir
StepHypRef Expression
1 pssss 3016 . . 3 (ABAB)
21orcd 639 . 2 (AB → (AB BA))
3 eqimss 2974 . . 3 (A = BAB)
43orcd 639 . 2 (A = B → (AB BA))
5 pssss 3016 . . 3 (BABA)
65olcd 640 . 2 (BA → (AB BA))
72, 4, 63jaoi 1184 1 ((AB A = B BA) → (AB BA))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 616   ∨ w3o 872   = wceq 1228   ⊆ wss 2894   ⊊ wpss 2895 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-3or 874  df-3an 875  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-in 2901  df-ss 2908  df-pss 2910 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator