Theorem nsspssun 3489

Description: Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)

Assertion

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

Proof of Theorem nsspssun

Step	Hyp	Ref	Expression
1		ssun2 3428	. . . 4 ⊢ B ⊆ (A ∪ B)
2	1	biantrur 492	. . 3 ⊢ (¬ (A ∪ B) ⊆ B ↔ (B ⊆ (A ∪ B) ∧ ¬ (A ∪ B) ⊆ B))
3		ssid 3291	. . . . 5 ⊢ B ⊆ B
4	3	biantru 491	. . . 4 ⊢ (A ⊆ B ↔ (A ⊆ B ∧ B ⊆ B))
5		unss 3438	. . . 4 ⊢ ((A ⊆ B ∧ B ⊆ B) ↔ (A ∪ B) ⊆ B)
6	4, 5	bitri 240	. . 3 ⊢ (A ⊆ B ↔ (A ∪ B) ⊆ B)
7	2, 6	xchnxbir 300	. 2 ⊢ (¬ A ⊆ B ↔ (B ⊆ (A ∪ B) ∧ ¬ (A ∪ B) ⊆ B))
8		dfpss3 3356	. 2 ⊢ (B ⊊ (A ∪ B) ↔ (B ⊆ (A ∪ B) ∧ ¬ (A ∪ B) ⊆ B))
9	7, 8	bitr4i 243	1 ⊢ (¬ A ⊆ B ↔ B ⊊ (A ∪ B))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ∪ cun 3208   ⊆ 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-un 3215  df-ss 3260  df-pss 3262

This theorem is referenced by:  disjpss  3602