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Theorem npss0 3260
Description: No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
npss0 ¬ A ⊊ ∅

Proof of Theorem npss0
StepHypRef Expression
1 0ss 3249 . . . 4 ∅ ⊆ A
21a1i 9 . . 3 (A ⊆ ∅ → ∅ ⊆ A)
3 imanim 784 . . 3 ((A ⊆ ∅ → ∅ ⊆ A) → ¬ (A ⊆ ∅ ¬ ∅ ⊆ A))
42, 3ax-mp 7 . 2 ¬ (A ⊆ ∅ ¬ ∅ ⊆ A)
5 dfpss3 3024 . 2 (A ⊊ ∅ ↔ (A ⊆ ∅ ¬ ∅ ⊆ A))
64, 5mtbir 595 1 ¬ A ⊊ ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wss 2911  wpss 2912  c0 3218
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-nul 3219
This theorem is referenced by: (None)
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