Theorem disjpss 4391

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 3939	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
2	1	biantru 529	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵))
3		ssin 4161	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵))
4	2, 3	bitri 274	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵))
5		sseq2 3943	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∩ 𝐵) ↔ 𝐵 ⊆ ∅))
6	4, 5	syl5bb 282	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ ∅))
7		ss0 4329	. . . . 5 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
8	6, 7	syl6bi 252	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ 𝐴 → 𝐵 = ∅))
9	8	necon3ad 2955	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ≠ ∅ → ¬ 𝐵 ⊆ 𝐴))
10	9	imp 406	. 2 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → ¬ 𝐵 ⊆ 𝐴)
11		nsspssun 4188	. . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐵 ∪ 𝐴))
12		uncom 4083	. . . 4 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
13	12	psseq2i 4021	. . 3 ⊢ (𝐴 ⊊ (𝐵 ∪ 𝐴) ↔ 𝐴 ⊊ (𝐴 ∪ 𝐵))
14	11, 13	bitri 274	. 2 ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∪ 𝐵))
15	10, 14	sylib 217	1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ≠ wne 2942   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883   ⊊ wpss 3884  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254

This theorem is referenced by:  omsucne  7706  isfin1-3  10073