Theorem nssinpss 3488

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 3476	. . 3 ⊢ (A ∩ B) ⊆ A
2	1	biantrur 492	. 2 ⊢ ((A ∩ B) ≠ A ↔ ((A ∩ B) ⊆ A ∧ (A ∩ B) ≠ A))
3		df-ss 3260	. . 3 ⊢ (A ⊆ B ↔ (A ∩ B) = A)
4	3	necon3bbii 2548	. 2 ⊢ (¬ A ⊆ B ↔ (A ∩ B) ≠ A)
5		df-pss 3262	. 2 ⊢ ((A ∩ B) ⊊ A ↔ ((A ∩ B) ⊆ A ∧ (A ∩ B) ≠ A))
6	2, 4, 5	3bitr4i 268	1 ⊢ (¬ A ⊆ B ↔ (A ∩ B) ⊊ A)

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ≠ wne 2517   ∩ cin 3209   ⊆ wss 3258   ⊊ wpss 3259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pss 3262

This theorem is referenced by: (None)