ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfpss2 GIF version

Theorem dfpss2 3057
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 2961 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
2 df-ne 2221 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
32anbi2i 438 . 2 ((𝐴𝐵𝐴𝐵) ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
41, 3bitri 177 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101  wb 102   = wceq 1259  wne 2220  wss 2945  wpss 2946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114  df-ne 2221  df-pss 2961
This theorem is referenced by:  dfpss3  3058  psstr  3077  sspsstr  3078  psssstr  3079  pssv  3295  disj4im  3304  ssnelpss  3317  onpsssuc  4323  f1imapss  5443
  Copyright terms: Public domain W3C validator