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Theorem difsnpssim 3477
 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 547 . 2 (A B → ¬ ¬ A B)
2 difss 3043 . . . 4 (B ∖ {A}) ⊆ B
32biantrur 287 . . 3 ((B ∖ {A}) ≠ B ↔ ((B ∖ {A}) ⊆ B (B ∖ {A}) ≠ B))
4 difsnb 3476 . . . 4 A B ↔ (B ∖ {A}) = B)
54necon3bbii 2216 . . 3 (¬ ¬ A B ↔ (B ∖ {A}) ≠ B)
6 df-pss 2906 . . 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 1370   ≠ wne 2182   ∖ cdif 2887   ⊆ wss 2890   ⊊ wpss 2891  {csn 3346 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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-v 2533  df-dif 2893  df-in 2897  df-ss 2904  df-pss 2906  df-sn 3352 This theorem is referenced by: (None)
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