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Theorem 0pss 3259
Description: The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)
Assertion
Ref Expression
0pss (∅ ⊊ AA ≠ ∅)

Proof of Theorem 0pss
StepHypRef Expression
1 0ss 3249 . . 3 ∅ ⊆ A
2 df-pss 2927 . . 3 (∅ ⊊ A ↔ (∅ ⊆ A ∅ ≠ A))
31, 2mpbiran 846 . 2 (∅ ⊊ A ↔ ∅ ≠ A)
4 necom 2283 . 2 (∅ ≠ AA ≠ ∅)
53, 4bitri 173 1 (∅ ⊊ AA ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wb 98  wne 2201  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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