Theorem sspsstrir 3046

Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by Jim Kingdon, 16-Jul-2018.)

Assertion

Ref	Expression
sspsstrir	|- ( ( A C. B \/ A = B \/ B C. A ) -> ( A C_ B \/ B C_ A ) )

Proof of Theorem sspsstrir

Step	Hyp	Ref	Expression
1		pssss 3039	. . 3 |- ( A C. B -> A C_ B )
2	1	orcd 652	. 2 |- ( A C. B -> ( A C_ B \/ B C_ A ) )
3		eqimss 2997	. . 3 |- ( A = B -> A C_ B )
4	3	orcd 652	. 2 |- ( A = B -> ( A C_ B \/ B C_ A ) )
5		pssss 3039	. . 3 |- ( B C. A -> B C_ A )
6	5	olcd 653	. 2 |- ( B C. A -> ( A C_ B \/ B C_ A ) )
7	2, 4, 6	3jaoi 1198	1 |- ( ( A C. B \/ A = B \/ B C. A ) -> ( A C_ B \/ B C_ A ) )

Syntax hints:   -> wi 4   \/ wo 629   \/ w3o 884   = wceq 1243   C_ wss 2917   C. 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)