Theorem nssinpss 1665

Description: Negation of subclass expressed in terms of intersection and proper subclass.

Assertion

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

Proof of Theorem nssinpss

Step	Hyp	Ref	Expression
1		inss1 1657	. . 3 ⊢ (A ∩ B) ⊆ A
2	1	biantrur 544	. 2 ⊢ (¬ A ⊆ (A ∩ B) ↔ ((A ∩ B) ⊆ A ∧ ¬ A ⊆ (A ∩ B)))
3		ssid 1519	. . . . 5 ⊢ A ⊆ A
4	3	biantrur 544	. . . 4 ⊢ (A ⊆ B ↔ (A ⊆ A ∧ A ⊆ B))
5		ssin 1659	. . . 4 ⊢ ((A ⊆ A ∧ A ⊆ B) ↔ A ⊆ (A ∩ B))
6	4, 5	bitr 151	. . 3 ⊢ (A ⊆ B ↔ A ⊆ (A ∩ B))
7	6	negbii 162	. 2 ⊢ (¬ A ⊆ B ↔ ¬ A ⊆ (A ∩ B))
8		dfpss3 1558	. 2 ⊢ ((A ∩ B) ⊂ A ↔ ((A ∩ B) ⊆ A ∧ ¬ A ⊆ (A ∩ B)))
9	2, 7, 8	3bitr4 158	1 ⊢ (¬ A ⊆ B ↔ (A ∩ B) ⊂ A)

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196   ∩ cin 1486   ⊆ wss 1487   ⊂ wpss 1488

This theorem is referenced by:  chrelat2 5758

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-v 1349  df-in 1491  df-ss 1492  df-pss 1494

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