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

Theorem difsnpssim 3532
 Description: (𝐵 ∖ {𝐴}) is a proper subclass of 𝐵 if 𝐴 is a member of 𝐵. In classical logic, the converse holds as well. (Contributed by Jim Kingdon, 9-Aug-2018.)
Assertion
Ref Expression
difsnpssim (𝐴𝐵 → (𝐵 ∖ {𝐴}) ⊊ 𝐵)

Proof of Theorem difsnpssim
StepHypRef Expression
1 notnot 567 . 2 (𝐴𝐵 → ¬ ¬ 𝐴𝐵)
2 difss 3095 . . . 4 (𝐵 ∖ {𝐴}) ⊆ 𝐵
32biantrur 291 . . 3 ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
4 difsnb 3531 . . . 4 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
54necon3bbii 2255 . . 3 (¬ ¬ 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵)
6 df-pss 2958 . . 3 ((𝐵 ∖ {𝐴}) ⊊ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
73, 5, 63bitr4i 205 . 2 (¬ ¬ 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
81, 7sylib 131 1 (𝐴𝐵 → (𝐵 ∖ {𝐴}) ⊊ 𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ∈ wcel 1407   ≠ wne 2218   ∖ cdif 2939   ⊆ wss 2942   ⊊ wpss 2943  {csn 3400 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036 This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-v 2574  df-dif 2945  df-in 2949  df-ss 2956  df-pss 2958  df-sn 3406 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator