Theorem npss0OLD 4015

Description: Obsolete proof of npss0 4014 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 3972	. . . 4 ⊢ ∅ ⊆ 𝐴
2	1	a1i 11	. . 3 ⊢ (𝐴 ⊆ ∅ → ∅ ⊆ 𝐴)
3		iman 440	. . 3 ⊢ ((𝐴 ⊆ ∅ → ∅ ⊆ 𝐴) ↔ ¬ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴))
4	2, 3	mpbi 220	. 2 ⊢ ¬ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴)
5		dfpss3 3693	. 2 ⊢ (𝐴 ⊊ ∅ ↔ (𝐴 ⊆ ∅ ∧ ¬ ∅ ⊆ 𝐴))
6	4, 5	mtbir 313	1 ⊢ ¬ 𝐴 ⊊ ∅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ⊆ wss 3574   ⊊ wpss 3575  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916

This theorem is referenced by: (None)