Theorem 0pss 4366

Description: The empty set is a proper subset of any nonempty set. Dual of pssv 4368. (Contributed by NM, 27-Feb-1996.)

Assertion

Ref	Expression
0pss	⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)

Proof of Theorem 0pss

Step	Hyp	Ref	Expression
1		0ss 4356	. . 3 ⊢ ∅ ⊆ 𝐴
2		df-pss 3924	. . 3 ⊢ (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
3	1, 2	mpbiran 721	. 2 ⊢ (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4		necom 3009	. 2 ⊢ (∅ ≠ 𝐴 ↔ 𝐴 ≠ ∅)
5	3, 4	bitri 278	1 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)

Syntax hints:   ↔ wb 209   ≠ wne 2956   ⊆ wss 3904   ⊊ wpss 3905  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-dif 3907  df-ss 3921  df-pss 3924  df-nul 4286

This theorem is referenced by:  php  9190  zornn0g  10488  prn0  10973  genpn0  10987  nqpr  10998  ltexprlem5  11024  reclem2pr  11032  suplem1pr  11036  alexsubALTlem4  24186  bj-2upln0  37625  bj-2upln1upl  37626  0pssin  44467