Theorem 0pss 4453

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 4406	. . 3 ⊢ ∅ ⊆ 𝐴
2		df-pss 3983	. . 3 ⊢ (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
3	1, 2	mpbiran 709	. 2 ⊢ (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4		necom 2992	. 2 ⊢ (∅ ≠ 𝐴 ↔ 𝐴 ≠ ∅)
5	3, 4	bitri 275	1 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)

Syntax hints:   ↔ wb 206   ≠ wne 2938   ⊆ wss 3963   ⊊ wpss 3964  ∅c0 4339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-dif 3966  df-ss 3980  df-pss 3983  df-nul 4340

This theorem is referenced by:  php  9245  phpOLD  9257  zornn0g  10543  prn0  11027  genpn0  11041  nqpr  11052  ltexprlem5  11078  reclem2pr  11086  suplem1pr  11090  alexsubALTlem4  24074  bj-2upln0  37006  bj-2upln1upl  37007  0pssin  43761