Theorem un00 4443

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 4159	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅))
2		un0 4391	. . 3 ⊢ (∅ ∪ ∅) = ∅
3	1, 2	eqtrdi 2789	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = ∅)
4		ssun1 4173	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
5		sseq2 4009	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐴 ⊆ ∅))
6	4, 5	mpbii 232	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 ⊆ ∅)
7		ss0b 4398	. . . 4 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
8	6, 7	sylib 217	. . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 = ∅)
9		ssun2 4174	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
10		sseq2 4009	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐵 ⊆ ∅))
11	9, 10	mpbii 232	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 ⊆ ∅)
12		ss0b 4398	. . . 4 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
13	11, 12	sylib 217	. . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 = ∅)
14	8, 13	jca 513	. 2 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))
15	3, 14	impbii 208	1 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅)

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∪ cun 3947   ⊆ wss 3949  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324

This theorem is referenced by:  undisj1  4462  undisj2  4463  disjpr2  4718  rankxplim3  9876  ssxr  11283  rpnnen2lem12  16168  wwlksnext  29147  asindmre  36571  tfsconcat00  42097  iunrelexp0  42453  uneqsn  42776