Theorem un00 4434

Description: Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)

Assertion

Ref	Expression
un00	⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅)

Proof of Theorem un00

Step	Hyp	Ref	Expression
1		uneq12 4150	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅))
2		un0 4382	. . 3 ⊢ (∅ ∪ ∅) = ∅
3	1, 2	eqtrdi 2780	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = ∅)
4		ssun1 4164	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
5		sseq2 4000	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐴 ⊆ ∅))
6	4, 5	mpbii 232	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 ⊆ ∅)
7		ss0b 4389	. . . 4 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
8	6, 7	sylib 217	. . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 = ∅)
9		ssun2 4165	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
10		sseq2 4000	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐵 ⊆ ∅))
11	9, 10	mpbii 232	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 ⊆ ∅)
12		ss0b 4389	. . . 4 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
13	11, 12	sylib 217	. . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 = ∅)
14	8, 13	jca 511	. 2 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))
15	3, 14	impbii 208	1 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅)

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∪ cun 3938   ⊆ wss 3940  ∅c0 4314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315

This theorem is referenced by:  undisj1  4453  undisj2  4454  disjpr2  4709  rankxplim3  9872  ssxr  11280  rpnnen2lem12  16165  wwlksnext  29616  asindmre  37061  tfsconcat00  42586  iunrelexp0  42942  uneqsn  43265