Theorem npss0 3589

Description: No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
npss0	⊢ ¬ A ⊊ ∅

Proof of Theorem npss0

Step	Hyp	Ref	Expression
1		0ss 3579	. . . 4 ⊢ ∅ ⊆ A
2	1	a1i 10	. . 3 ⊢ (A ⊆ ∅ → ∅ ⊆ A)
3		iman 413	. . 3 ⊢ ((A ⊆ ∅ → ∅ ⊆ A) ↔ ¬ (A ⊆ ∅ ∧ ¬ ∅ ⊆ A))
4	2, 3	mpbi 199	. 2 ⊢ ¬ (A ⊆ ∅ ∧ ¬ ∅ ⊆ A)
5		dfpss3 3355	. 2 ⊢ (A ⊊ ∅ ↔ (A ⊆ ∅ ∧ ¬ ∅ ⊆ A))
6	4, 5	mtbir 290	1 ⊢ ¬ A ⊊ ∅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ⊆ wss 3257   ⊊ wpss 3258  ∅c0 3550

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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-pss 3261  df-nul 3551

This theorem is referenced by: (None)