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Theorem sspss 3690
Description: Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)
Assertion
Ref Expression
sspss (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem sspss
StepHypRef Expression
1 dfpss2 3676 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
21simplbi2 654 . . . 4 (𝐴𝐵 → (¬ 𝐴 = 𝐵𝐴𝐵))
32con1d 139 . . 3 (𝐴𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))
43orrd 393 . 2 (𝐴𝐵 → (𝐴𝐵𝐴 = 𝐵))
5 pssss 3686 . . 3 (𝐴𝐵𝐴𝐵)
6 eqimss 3642 . . 3 (𝐴 = 𝐵𝐴𝐵)
75, 6jaoi 394 . 2 ((𝐴𝐵𝐴 = 𝐵) → 𝐴𝐵)
84, 7impbii 199 1 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 383   = wceq 1480  wss 3560  wpss 3561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ne 2791  df-in 3567  df-ss 3574  df-pss 3576
This theorem is referenced by:  sspsstri  3693  sspsstr  3696  psssstr  3697  ordsseleq  5721  sorpssuni  6911  sorpssint  6912  ssnnfi  8139  ackbij1b  9021  fin23lem40  9133  zorng  9286  psslinpr  9813  suplem2pr  9835  ressval3d  15877  mrissmrcd  16240  pgpssslw  17969  pgpfac1lem5  18418  idnghm  22487  dfon2lem4  31445  finminlem  32007  lkrss2N  33975  dvh3dim3N  36257
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