Theorem 0pss 4413

Description: The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.)

Assertion

Ref	Expression
0pss	⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)

Proof of Theorem 0pss

Step	Hyp	Ref	Expression
1		0ss 4366	. . 3 ⊢ ∅ ⊆ 𝐴
2		df-pss 3937	. . 3 ⊢ (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
3	1, 2	mpbiran 709	. 2 ⊢ (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4		necom 2979	. 2 ⊢ (∅ ≠ 𝐴 ↔ 𝐴 ≠ ∅)
5	3, 4	bitri 275	1 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)

Syntax hints:   ↔ wb 206   ≠ wne 2926   ⊆ wss 3917   ⊊ wpss 3918  ∅c0 4299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-dif 3920  df-ss 3934  df-pss 3937  df-nul 4300

This theorem is referenced by:  php  9177  zornn0g  10465  prn0  10949  genpn0  10963  nqpr  10974  ltexprlem5  11000  reclem2pr  11008  suplem1pr  11012  alexsubALTlem4  23944  bj-2upln0  37018  bj-2upln1upl  37019  0pssin  43767