Theorem dfpss3 2130

Description: Alternate definition of proper subclass.

Assertion

Ref	Expression
dfpss3	|- (A (. B <-> (A (_ B /\ -. B (_ A))

Proof of Theorem dfpss3

Step	Hyp	Ref	Expression
1		eqss 2073	. . . 4 |- (A = B <-> (A (_ B /\ B (_ A))
2	1	negbii 187	. . 3 |- (-. A = B <-> -. (A (_ B /\ B (_ A))
3	2	anbi2i 480	. 2 |- ((A (_ B /\ -. A = B) <-> (A (_ B /\ -. (A (_ B /\ B (_ A)))
4		dfpss2 2129	. 2 |- (A (. B <-> (A (_ B /\ -. A = B))
5		anclb 329	. . . 4 |- ((A (_ B -> B (_ A) <-> (A (_ B -> (A (_ B /\ B (_ A)))
6		iman 237	. . . 4 |- ((A (_ B -> B (_ A) <-> -. (A (_ B /\ -. B (_ A))
7		iman 237	. . . 4 |- ((A (_ B -> (A (_ B /\ B (_ A)) <-> -. (A (_ B /\ -. (A (_ B /\ B (_ A)))
8	5, 6, 7	3bitr3 181	. . 3 |- (-. (A (_ B /\ -. B (_ A) <-> -. (A (_ B /\ -. (A (_ B /\ B (_ A)))
9	8	con4bii 522	. 2 |- ((A (_ B /\ -. B (_ A) <-> (A (_ B /\ -. (A (_ B /\ B (_ A)))
10	3, 4, 9	3bitr4 183	1 |- (A (. B <-> (A (_ B /\ -. B (_ A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   (_ wss 2043   (. wpss 2044

This theorem is referenced by:  pssirr 2142  pssn2lp 2143  nssinpss 2236  nsspssun 2237  php3 4501  prlem934 5119  reclem2pr 5137  chpsscon3t 9364  chpssat 10227

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-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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