Theorem sspss 2145

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 2133	. . . . . 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 2143	. . 3 |- (A (. B -> A (_ B)
7		eqimss 2109	. . 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 956   (_ wss 2047   (. wpss 2048

This theorem is referenced by:  sspsstri 2148  ssnpss 2149  sspsstr 2151  psssstr 2152  ssnnfi 4535  ssnnfiOLD 4536  zorn 4797  psslinpr 5135  suplem2pr 5162

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-in 2051  df-ss 2053  df-pss 2055

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