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Theorem 0pss 3588
 Description: The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.)
Assertion
Ref Expression
0pss (AA)

Proof of Theorem 0pss
StepHypRef Expression
1 0ss 3579 . . 3 A
2 df-pss 3261 . . 3 (A ↔ ( A A))
31, 2mpbiran 884 . 2 (AA)
4 necom 2597 . 2 (AA)
53, 4bitri 240 1 (AA)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ≠ wne 2516   ⊆ wss 3257   ⊊ wpss 3258  ∅c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-pss 3261  df-nul 3551 This theorem is referenced by: (None)
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