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Theorem npss0OLD 3987
 Description: Obsolete proof of npss0 3986 as of 14-Jul-2021. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
npss0OLD ¬ 𝐴 ⊊ ∅

Proof of Theorem npss0OLD
StepHypRef Expression
1 0ss 3944 . . . 4 ∅ ⊆ 𝐴
21a1i 11 . . 3 (𝐴 ⊆ ∅ → ∅ ⊆ 𝐴)
3 iman 440 . . 3 ((𝐴 ⊆ ∅ → ∅ ⊆ 𝐴) ↔ ¬ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴))
42, 3mpbi 220 . 2 ¬ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴)
5 dfpss3 3671 . 2 (𝐴 ⊊ ∅ ↔ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴))
64, 5mtbir 313 1 ¬ 𝐴 ⊊ ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ⊆ wss 3555   ⊊ wpss 3556  ∅c0 3891 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892 This theorem is referenced by: (None)
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