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Theorem sspsstrir 3040
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 3033 . . 3 (ABAB)
21orcd 651 . 2 (AB → (AB BA))
3 eqimss 2991 . . 3 (A = BAB)
43orcd 651 . 2 (A = B → (AB BA))
5 pssss 3033 . . 3 (BABA)
65olcd 652 . 2 (BA → (AB BA))
72, 4, 63jaoi 1197 1 ((AB A = B BA) → (AB BA))
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628   w3o 883   = wceq 1242  wss 2911  wpss 2912
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-pss 2927
This theorem is referenced by: (None)
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