Theorem disjpss 3601

Description: A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)

Assertion

Ref	Expression
disjpss	⊢ (((A ∩ B) = ∅ ∧ B ≠ ∅) → A ⊊ (A ∪ B))

Proof of Theorem disjpss

Step	Hyp	Ref	Expression
1		ssid 3290	. . . . . . . 8 ⊢ B ⊆ B
2	1	biantru 491	. . . . . . 7 ⊢ (B ⊆ A ↔ (B ⊆ A ∧ B ⊆ B))
3		ssin 3477	. . . . . . 7 ⊢ ((B ⊆ A ∧ B ⊆ B) ↔ B ⊆ (A ∩ B))
4	2, 3	bitri 240	. . . . . 6 ⊢ (B ⊆ A ↔ B ⊆ (A ∩ B))
5		sseq2 3293	. . . . . 6 ⊢ ((A ∩ B) = ∅ → (B ⊆ (A ∩ B) ↔ B ⊆ ∅))
6	4, 5	syl5bb 248	. . . . 5 ⊢ ((A ∩ B) = ∅ → (B ⊆ A ↔ B ⊆ ∅))
7		ss0 3581	. . . . 5 ⊢ (B ⊆ ∅ → B = ∅)
8	6, 7	syl6bi 219	. . . 4 ⊢ ((A ∩ B) = ∅ → (B ⊆ A → B = ∅))
9	8	necon3ad 2552	. . 3 ⊢ ((A ∩ B) = ∅ → (B ≠ ∅ → ¬ B ⊆ A))
10	9	imp 418	. 2 ⊢ (((A ∩ B) = ∅ ∧ B ≠ ∅) → ¬ B ⊆ A)
11		nsspssun 3488	. . 3 ⊢ (¬ B ⊆ A ↔ A ⊊ (B ∪ A))
12		uncom 3408	. . . 4 ⊢ (B ∪ A) = (A ∪ B)
13	12	psseq2i 3359	. . 3 ⊢ (A ⊊ (B ∪ A) ↔ A ⊊ (A ∪ B))
14	11, 13	bitri 240	. 2 ⊢ (¬ B ⊆ A ↔ A ⊊ (A ∪ B))
15	10, 14	sylib 188	1 ⊢ (((A ∩ B) = ∅ ∧ B ≠ ∅) → A ⊊ (A ∪ B))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2516   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257   ⊊ wpss 3258  ∅c0 3550

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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-pss 3261  df-nul 3551

This theorem is referenced by: (None)