Theorem 0pss 4404

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 4357	. . 3 ⊢ ∅ ⊆ 𝐴
2		df-pss 3927	. . 3 ⊢ (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
3	1, 2	mpbiran 721	. 2 ⊢ (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4		necom 3013	. 2 ⊢ (∅ ≠ 𝐴 ↔ 𝐴 ≠ ∅)
5	3, 4	bitri 278	1 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)

Syntax hints:   ↔ wb 209   ≠ wne 2960   ⊆ wss 3907   ⊊ wpss 3908  ∅c0 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-dif 3910  df-ss 3924  df-pss 3927  df-nul 4289

This theorem is referenced by:  php  9179  zornn0g  10477  prn0  10962  genpn0  10976  nqpr  10987  ltexprlem5  11013  reclem2pr  11021  suplem1pr  11025  alexsubALTlem4  24168  bj-2upln0  37520  bj-2upln1upl  37521  0pssin  44359