Theorem disjpss 4172

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 3773	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
2	1	biantru 519	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵))
3		ssin 3983	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵))
4	2, 3	bitri 264	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵))
5		sseq2 3776	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∩ 𝐵) ↔ 𝐵 ⊆ ∅))
6	4, 5	syl5bb 272	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ ∅))
7		ss0 4119	. . . . 5 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
8	6, 7	syl6bi 243	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ 𝐴 → 𝐵 = ∅))
9	8	necon3ad 2956	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ≠ ∅ → ¬ 𝐵 ⊆ 𝐴))
10	9	imp 393	. 2 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → ¬ 𝐵 ⊆ 𝐴)
11		nsspssun 4006	. . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐵 ∪ 𝐴))
12		uncom 3908	. . . 4 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
13	12	psseq2i 3847	. . 3 ⊢ (𝐴 ⊊ (𝐵 ∪ 𝐴) ↔ 𝐴 ⊊ (𝐴 ∪ 𝐵))
14	11, 13	bitri 264	. 2 ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∪ 𝐵))
15	10, 14	sylib 208	1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   = wceq 1631   ≠ wne 2943   ∪ cun 3721   ∩ cin 3722   ⊆ wss 3723   ⊊ wpss 3724  ∅c0 4063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064

This theorem is referenced by:  isfin1-3  9414