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Theorem dfpss2 3029
 Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
Assertion
Ref Expression
dfpss2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))

Proof of Theorem dfpss2
StepHypRef Expression
1 df-pss 2933 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
2 df-ne 2206 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
32anbi2i 430 . 2 ((𝐴𝐵𝐴𝐵) ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
41, 3bitri 173 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   = wceq 1243   ≠ wne 2204   ⊆ wss 2917   ⊊ wpss 2918 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101 This theorem depends on definitions:  df-bi 110  df-ne 2206  df-pss 2933 This theorem is referenced by:  dfpss3  3030  psstr  3049  sspsstr  3050  psssstr  3051  pssv  3267  disj4im  3276  ssnelpss  3289  onpsssuc  4295  f1imapss  5415
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