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Theorem sspss 3368
 Description: Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)
Assertion
Ref Expression
sspss (A B ↔ (AB A = B))

Proof of Theorem sspss
StepHypRef Expression
1 dfpss2 3354 . . . . 5 (AB ↔ (A B ¬ A = B))
21simplbi2 608 . . . 4 (A B → (¬ A = BAB))
32con1d 116 . . 3 (A B → (¬ ABA = B))
43orrd 367 . 2 (A B → (AB A = B))
5 pssss 3364 . . 3 (ABA B)
6 eqimss 3323 . . 3 (A = BA B)
75, 6jaoi 368 . 2 ((AB A = B) → A B)
84, 7impbii 180 1 (A B ↔ (AB A = B))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   = wceq 1642   ⊆ wss 3257   ⊊ wpss 3258 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pss 3261 This theorem is referenced by:  sspsstri  3371  sspsstr  3374  psssstr  3375  ssfin  4470
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