Theorem nssinpss 4247

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 4217	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	biantrur 530	. 2 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴))
3		dfss2 3949	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
4	3	necon3bbii 2980	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ≠ 𝐴)
5		df-pss 3951	. 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 206   ∧ wa 395   ≠ wne 2933   ∩ cin 3930   ⊆ wss 3931   ⊊ wpss 3932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-v 3466  df-in 3938  df-ss 3948  df-pss 3951

This theorem is referenced by:  fbfinnfr  23784  chrelat2i  32351