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Theorem npss 3695
Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3598. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
npss 𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem npss
StepHypRef Expression
1 pm4.61 442 . . 3 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
2 dfpss2 3670 . . 3 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
31, 2bitr4i 267 . 2 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ 𝐴𝐵)
43con1bii 346 1 𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wss 3555  wpss 3556
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 386  df-ne 2791  df-pss 3571
This theorem is referenced by:  ttukeylem7  9281  canthp1lem2  9419  pgpfac1lem1  18394  lspsncv0  19065  obslbs  19993
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