Theorem dfpss3 2145

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 2088	. . . 4 |- (A = B <-> (A (_ B /\ B (_ A))
2	1	notbii 187	. . 3 |- (-. A = B <-> -. (A (_ B /\ B (_ A))
3	2	anbi2i 483	. 2 |- ((A (_ B /\ -. A = B) <-> (A (_ B /\ -. (A (_ B /\ B (_ A)))
4		dfpss2 2144	. 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	3bitr3i 181	. . 3 |- (-. (A (_ B /\ -. B (_ A) <-> -. (A (_ B /\ -. (A (_ B /\ B (_ A)))
9	8	con4bii 526	. 2 |- ((A (_ B /\ -. B (_ A) <-> (A (_ B /\ -. (A (_ B /\ B (_ A)))
10	3, 4, 9	3bitr4i 183	1 |- (A (. B <-> (A (_ B /\ -. B (_ A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 960   (_ wss 2058   (. wpss 2059

This theorem is referenced by:  pssirr 2157  pssn2lp 2158  nssinpss 2251  nsspssun 2252  php3 4535  prlem934 5159  reclem2pr 5177  chpsscon3 9450  chpssati 10315  vxveqv 10497

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-in 2062  df-ss 2064  df-pss 2066

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