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Theorem difsnpssim 3498
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 (A B → (B ∖ {A}) ⊊ B)

Proof of Theorem difsnpssim
StepHypRef Expression
1 notnot1 559 . 2 (A B → ¬ ¬ A B)
2 difss 3064 . . . 4 (B ∖ {A}) ⊆ B
32biantrur 287 . . 3 ((B ∖ {A}) ≠ B ↔ ((B ∖ {A}) ⊆ B (B ∖ {A}) ≠ B))
4 difsnb 3497 . . . 4 A B ↔ (B ∖ {A}) = B)
54necon3bbii 2236 . . 3 (¬ ¬ A B ↔ (B ∖ {A}) ≠ B)
6 df-pss 2927 . . 3 ((B ∖ {A}) ⊊ B ↔ ((B ∖ {A}) ⊆ B (B ∖ {A}) ≠ B))
73, 5, 63bitr4i 201 . 2 (¬ ¬ A B ↔ (B ∖ {A}) ⊊ B)
81, 7sylib 127 1 (A B → (B ∖ {A}) ⊊ B)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wcel 1390  wne 2201  cdif 2908  wss 2911  wpss 2912  {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-pss 2927  df-sn 3373
This theorem is referenced by: (None)
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