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Theorem npss 4062
Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3957. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
npss 𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem npss
StepHypRef Expression
1 pm4.61 408 . . 3 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
2 dfpss2 4037 . . 3 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
31, 2bitr4i 281 . 2 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ 𝐴𝐵)
43con1bii 360 1 𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wss 3908  wpss 3909
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 210  df-an 400  df-ne 3012  df-pss 3927
This theorem is referenced by:  ttukeylem7  9926  canthp1lem2  10064  pgpfac1lem1  19187  lspsncv0  19909  obslbs  20417  ssmxidl  31021  fvineqsneq  34790
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