Theorem 0pss 4410

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 4363	. . 3 ⊢ ∅ ⊆ 𝐴
2		df-pss 3934	. . 3 ⊢ (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
3	1, 2	mpbiran 709	. 2 ⊢ (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4		necom 2978	. 2 ⊢ (∅ ≠ 𝐴 ↔ 𝐴 ≠ ∅)
5	3, 4	bitri 275	1 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)

Syntax hints:   ↔ wb 206   ≠ wne 2925   ⊆ wss 3914   ⊊ wpss 3915  ∅c0 4296

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-dif 3917  df-ss 3931  df-pss 3934  df-nul 4297

This theorem is referenced by:  php  9171  zornn0g  10458  prn0  10942  genpn0  10956  nqpr  10967  ltexprlem5  10993  reclem2pr  11001  suplem1pr  11005  alexsubALTlem4  23937  bj-2upln0  37011  bj-2upln1upl  37012  0pssin  43760