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Theorem npss 3914
Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3813. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
npss 𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem npss
StepHypRef Expression
1 pm4.61 394 . . 3 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
2 dfpss2 3889 . . 3 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
31, 2bitr4i 270 . 2 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ 𝐴𝐵)
43con1bii 348 1 𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wss 3769  wpss 3770
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 199  df-an 386  df-ne 2972  df-pss 3785
This theorem is referenced by:  ttukeylem7  9625  canthp1lem2  9763  pgpfac1lem1  18789  lspsncv0  19468  lspsncv0OLD  19469  obslbs  20399
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