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Theorem npss 3851
Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3751. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
npss 𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem npss
StepHypRef Expression
1 pm4.61 441 . . 3 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
2 dfpss2 3826 . . 3 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
31, 2bitr4i 267 . 2 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ 𝐴𝐵)
43con1bii 345 1 𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1624  wss 3707  wpss 3708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-an 385  df-ne 2925  df-pss 3723
This theorem is referenced by:  ttukeylem7  9521  canthp1lem2  9659  pgpfac1lem1  18665  lspsncv0  19340  obslbs  20268
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