Theorem nsspssun 1666

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

Assertion

Ref	Expression
nsspssun	⊢ (¬ A ⊆ B ↔ B ⊂ (A ∪ B))

Proof of Theorem nsspssun

Step	Hyp	Ref	Expression
1		ssun2 1622	. . 3 ⊢ B ⊆ (A ∪ B)
2	1	biantrur 544	. 2 ⊢ (¬ (A ∪ B) ⊆ B ↔ (B ⊆ (A ∪ B) ∧ ¬ (A ∪ B) ⊆ B))
3		ssid 1519	. . . . 5 ⊢ B ⊆ B
4	3	biantru 543	. . . 4 ⊢ (A ⊆ B ↔ (A ⊆ B ∧ B ⊆ B))
5		unss 1632	. . . 4 ⊢ ((A ⊆ B ∧ B ⊆ B) ↔ (A ∪ B) ⊆ B)
6	4, 5	bitr 151	. . 3 ⊢ (A ⊆ B ↔ (A ∪ B) ⊆ B)
7	6	negbii 162	. 2 ⊢ (¬ A ⊆ B ↔ ¬ (A ∪ B) ⊆ B)
8		dfpss3 1558	. 2 ⊢ (B ⊂ (A ∪ B) ↔ (B ⊆ (A ∪ B) ∧ ¬ (A ∪ B) ⊆ B))
9	2, 7, 8	3bitr4 158	1 ⊢ (¬ A ⊆ B ↔ B ⊂ (A ∪ B))

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196   ∪ cun 1485   ⊆ wss 1487   ⊂ wpss 1488

This theorem is referenced by:  disjpss 1738

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-or 197  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-un 1490  df-in 1491  df-ss 1492  df-pss 1494

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