Theorem sspss 2135

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 2123	. . . . . 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 2133	. . 3 |- (A (. B -> A (_ B)
7		eqimss 2099	. . 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 953   (_ wss 2037   (. wpss 2038

This theorem is referenced by:  sspsstri 2138  ssnpss 2139  sspsstr 2141  psssstr 2142  ssnn 4514  zorn 4769  psslinpr 5107  suplem2pr 5134

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-in 2041  df-ss 2043  df-pss 2045

Copyright terms: Public domain