Theorem nssinpss 4257

Description: Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
nssinpss	⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴)

Proof of Theorem nssinpss

Step	Hyp	Ref	Expression
1		inss1 4229	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	biantrur 530	. 2 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴))
3		df-ss 3964	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
4	3	necon3bbii 2985	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ≠ 𝐴)
5		df-pss 3966	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴))
6	2, 4, 5	3bitr4i 303	1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ≠ wne 2937   ∩ cin 3946   ⊆ wss 3947   ⊊ wpss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-v 3473  df-in 3954  df-ss 3964  df-pss 3966

This theorem is referenced by:  fbfinnfr  23744  chrelat2i  32174