Theorem sspsstri 2711

Description: Two ways of stating trichotomy with respect to inclusion.

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		sspss 2707	. . 3 |- (A C_ B <-> (A C. B \/ A = B))
2		sspss 2707	. . . 4 |- (B C_ A <-> (B C. A \/ B = A))
3		eqcom 1886	. . . . 5 |- (B = A <-> A = B)
4	3	orbi2i 275	. . . 4 |- ((B C. A \/ B = A) <-> (B C. A \/ A = B))
5	2, 4	bitri 190	. . 3 |- (B C_ A <-> (B C. A \/ A = B))
6	1, 5	orbi12i 277	. 2 |- ((A C_ B \/ B C_ A) <-> ((A C. B \/ A = B) \/ (B C. A \/ A = B)))
7		orordir 289	. 2 |- (((A C. B \/ B C. A) \/ A = B) <-> ((A C. B \/ A = B) \/ (B C. A \/ A = B)))
8		or23 284	. . 3 |- (((A C. B \/ B C. A) \/ A = B) <-> ((A C. B \/ A = B) \/ B C. A))
9		df-3or 859	. . 3 |- ((A C. B \/ A = B \/ B C. A) <-> ((A C. B \/ A = B) \/ B C. A))
10	8, 9	bitr4i 193	. 2 |- (((A C. B \/ B C. A) \/ A = B) <-> (A C. B \/ A = B \/ B C. A))
11	6, 7, 10	3bitr2i 196	1 |- ((A C_ B \/ B C_ A) <-> (A C. B \/ A = B \/ B C. A))

Syntax hints:   <-> wb 163   \/ wo 239   \/ w3o 857   = wceq 1298   C_ wss 2593   C. wpss 2594

This theorem is referenced by:  ordtri3or 3691  zorn 5959  funpsstri 13835

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-in 2603  df-ss 2605  df-pss 2607

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