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Theorem sspssn 3048
 Description: Like pssn2lp 3045 but for subset and proper subset. (Contributed by Jim Kingdon, 17-Jul-2018.)
Assertion
Ref Expression
sspssn ¬ (𝐴𝐵𝐵𝐴)

Proof of Theorem sspssn
StepHypRef Expression
1 pm3.24 627 . 2 ¬ (𝐵𝐴 ∧ ¬ 𝐵𝐴)
2 ssnpss 3047 . . . 4 (𝐴𝐵 → ¬ 𝐵𝐴)
32anim2i 324 . . 3 ((𝐵𝐴𝐴𝐵) → (𝐵𝐴 ∧ ¬ 𝐵𝐴))
43ancoms 255 . 2 ((𝐴𝐵𝐵𝐴) → (𝐵𝐴 ∧ ¬ 𝐵𝐴))
51, 4mto 588 1 ¬ (𝐴𝐵𝐵𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ⊆ 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  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933 This theorem is referenced by:  sspsstr  3050  psssstr  3051
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