Theorem sspss 2141

Description: Subclass in terms of proper subclass.

Assertion

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

Proof of Theorem sspss

Step	Hyp	Ref	Expression
1		dfpss2 2129	. . . . . 6 |- (A (. B <-> (A (_ B /\ -. A = B))
2	1	biimpr 152	. . . . 5 |- ((A (_ B /\ -. A = B) -> A (. B)
3	2	ex 373	. . . 4 |- (A (_ B -> (-. A = B -> A (. B))
4	3	con1d 93	. . 3 |- (A (_ B -> (-. A (. B -> A = B))
5	4	orrd 233	. 2 |- (A (_ B -> (A (. B \/ A = B))
6		pssss 2139	. . 3 |- (A (. B -> A (_ B)
7		eqimss 2105	. . 3 |- (A = B -> A (_ B)
8	6, 7	jaoi 341	. 2 |- ((A (. B \/ A = B) -> A (_ B)
9	5, 8	impbi 157	1 |- (A (_ B <-> (A (. B \/ A = B))

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 954   (_ wss 2043   (. wpss 2044

This theorem is referenced by:  sspsstri 2144  ssnpss 2145  sspsstr 2147  psssstr 2148  ssnn 4520  zorn 4777  psslinpr 5115  suplem2pr 5142

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-in 2047  df-ss 2049  df-pss 2051

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