Theorem 0pss 3589

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 3580	. . 3 ⊢ ∅ ⊆ A
2		df-pss 3262	. . 3 ⊢ (∅ ⊊ A ↔ (∅ ⊆ A ∧ ∅ ≠ A))
3	1, 2	mpbiran 884	. 2 ⊢ (∅ ⊊ A ↔ ∅ ≠ A)
4		necom 2598	. 2 ⊢ (∅ ≠ A ↔ A ≠ ∅)
5	3, 4	bitri 240	1 ⊢ (∅ ⊊ A ↔ A ≠ ∅)

Syntax hints:   ↔ wb 176   ≠ wne 2517   ⊆ wss 3258   ⊊ wpss 3259  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-pss 3262  df-nul 3552

This theorem is referenced by: (None)