Theorem sspsstri 3709

Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)

Assertion

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

Proof of Theorem sspsstri

Step	Hyp	Ref	Expression
1		or32 549	. 2 |- ( ( ( A C. B \/ B C. A ) \/ A = B ) <-> ( ( A C. B \/ A = B ) \/ B C. A ) )
2		sspss 3706	. . . 4 |- ( A C_ B <-> ( A C. B \/ A = B ) )
3		sspss 3706	. . . . 5 |- ( B C_ A <-> ( B C. A \/ B = A ) )
4		eqcom 2629	. . . . . 6 |- ( B = A <-> A = B )
5	4	orbi2i 541	. . . . 5 |- ( ( B C. A \/ B = A ) <-> ( B C. A \/ A = B ) )
6	3, 5	bitri 264	. . . 4 |- ( B C_ A <-> ( B C. A \/ A = B ) )
7	2, 6	orbi12i 543	. . 3 |- ( ( A C_ B \/ B C_ A ) <-> ( ( A C. B \/ A = B ) \/ ( B C. A \/ A = B ) ) )
8		orordir 553	. . 3 |- ( ( ( A C. B \/ B C. A ) \/ A = B ) <-> ( ( A C. B \/ A = B ) \/ ( B C. A \/ A = B ) ) )
9	7, 8	bitr4i 267	. 2 |- ( ( A C_ B \/ B C_ A ) <-> ( ( A C. B \/ B C. A ) \/ A = B ) )
10		df-3or 1038	. 2 |- ( ( A C. B \/ A = B \/ B C. A ) <-> ( ( A C. B \/ A = B ) \/ B C. A ) )
11	1, 9, 10	3bitr4i 292	1 |- ( ( A C_ B \/ B C_ A ) <-> ( A C. B \/ A = B \/ B C. A ) )

Syntax hints:   <-> wb 196   \/ wo 383   \/ w3o 1036   = wceq 1483   C_ wss 3574   C. wpss 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-in 3581  df-ss 3588  df-pss 3590

This theorem is referenced by:  ordtri3or  5755  sorpss  6942  sorpssi  6943  funpsstri  31663