Theorem disjpss 3980

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 3587	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
2	1	biantru 525	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵))
3		ssin 3797	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵))
4	2, 3	bitri 263	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵))
5		sseq2 3590	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∩ 𝐵) ↔ 𝐵 ⊆ ∅))
6	4, 5	syl5bb 271	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ ∅))
7		ss0 3926	. . . . 5 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
8	6, 7	syl6bi 242	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ 𝐴 → 𝐵 = ∅))
9	8	necon3ad 2795	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ≠ ∅ → ¬ 𝐵 ⊆ 𝐴))
10	9	imp 444	. 2 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → ¬ 𝐵 ⊆ 𝐴)
11		nsspssun 3819	. . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐵 ∪ 𝐴))
12		uncom 3719	. . . 4 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
13	12	psseq2i 3659	. . 3 ⊢ (𝐴 ⊊ (𝐵 ∪ 𝐴) ↔ 𝐴 ⊊ (𝐴 ∪ 𝐵))
14	11, 13	bitri 263	. 2 ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∪ 𝐵))
15	10, 14	sylib 207	1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ≠ wne 2780   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875

This theorem is referenced by:  isfin1-3  9091