Theorem npss0OLD 3967

Description: Obsolete proof of npss0 3966 as of 14-Jul-2021. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
npss0OLD	⊢ ¬ 𝐴 ⊊ ∅

Proof of Theorem npss0OLD

Step	Hyp	Ref	Expression
1		0ss 3924	. . . 4 ⊢ ∅ ⊆ 𝐴
2	1	a1i 11	. . 3 ⊢ (𝐴 ⊆ ∅ → ∅ ⊆ 𝐴)
3		iman 439	. . 3 ⊢ ((𝐴 ⊆ ∅ → ∅ ⊆ 𝐴) ↔ ¬ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴))
4	2, 3	mpbi 219	. 2 ⊢ ¬ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴)
5		dfpss3 3655	. 2 ⊢ (𝐴 ⊊ ∅ ↔ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴))
6	4, 5	mtbir 312	1 ⊢ ¬ 𝐴 ⊊ ∅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875

This theorem is referenced by: (None)