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Theorem sspsstrir 3046
 Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by Jim Kingdon, 16-Jul-2018.)
Assertion
Ref Expression
sspsstrir ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴𝐵𝐵𝐴))

Proof of Theorem sspsstrir
StepHypRef Expression
1 pssss 3039 . . 3 (𝐴𝐵𝐴𝐵)
21orcd 652 . 2 (𝐴𝐵 → (𝐴𝐵𝐵𝐴))
3 eqimss 2997 . . 3 (𝐴 = 𝐵𝐴𝐵)
43orcd 652 . 2 (𝐴 = 𝐵 → (𝐴𝐵𝐵𝐴))
5 pssss 3039 . . 3 (𝐵𝐴𝐵𝐴)
65olcd 653 . 2 (𝐵𝐴 → (𝐴𝐵𝐵𝐴))
72, 4, 63jaoi 1198 1 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴𝐵𝐵𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 629   ∨ w3o 884   = wceq 1243   ⊆ wss 2917   ⊊ wpss 2918 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-pss 2933 This theorem is referenced by: (None)
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