Theorem sspsstri 2146

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

Assertion

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

Proof of Theorem sspsstri

Step	Hyp	Ref	Expression
1		sspss 2143	. . 3 |- (A (_ B <-> (A (. B \/ A = B))
2		sspss 2143	. . . 4 |- (B (_ A <-> (B (. A \/ B = A))
3		eqcom 1476	. . . . 5 |- (B = A <-> A = B)
4	3	orbi2i 255	. . . 4 |- ((B (. A \/ B = A) <-> (B (. A \/ A = B))
5	2, 4	bitr 173	. . 3 |- (B (_ A <-> (B (. A \/ A = B))
6	1, 5	orbi12i 257	. 2 |- ((A (_ B \/ B (_ A) <-> ((A (. B \/ A = B) \/ (B (. A \/ A = B)))
7		orordir 267	. 2 |- (((A (. B \/ B (. A) \/ A = B) <-> ((A (. B \/ A = B) \/ (B (. A \/ A = B)))
8		or23 263	. . 3 |- (((A (. B \/ B (. A) \/ A = B) <-> ((A (. B \/ A = B) \/ B (. A))
9		df-3or 775	. . 3 |- ((A (. B \/ A = B \/ B (. A) <-> ((A (. B \/ A = B) \/ B (. A))
10	8, 9	bitr4 176	. 2 |- (((A (. B \/ B (. A) \/ A = B) <-> (A (. B \/ A = B \/ B (. A))
11	6, 7, 10	3bitr2 179	1 |- ((A (_ B \/ B (_ A) <-> (A (. B \/ A = B \/ B (. A))

Syntax hints:   <-> wb 146   \/ wo 222   \/ w3o 773   = wceq 955   (_ wss 2045   (. wpss 2046

This theorem is referenced by:  zorn 4784

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-in 2049  df-ss 2051  df-pss 2053

Copyright terms: Public domain