Theorem npss0 3266

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 3255	. . . 4 ⊢ ∅ ⊆ 𝐴
2	1	a1i 9	. . 3 ⊢ (𝐴 ⊆ ∅ → ∅ ⊆ 𝐴)
3		imanim 785	. . 3 ⊢ ((𝐴 ⊆ ∅ → ∅ ⊆ 𝐴) → ¬ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴))
4	2, 3	ax-mp 7	. 2 ⊢ ¬ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴)
5		dfpss3 3030	. 2 ⊢ (𝐴 ⊊ ∅ ↔ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴))
6	4, 5	mtbir 596	1 ⊢ ¬ 𝐴 ⊊ ∅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ⊆ wss 2917   ⊊ wpss 2918  ∅c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-pss 2933  df-nul 3225

This theorem is referenced by: (None)