Theorem un00 3538

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 3353	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅))
2		un0 3525	. . 3 ⊢ (∅ ∪ ∅) = ∅
3	1, 2	eqtrdi 2278	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = ∅)
4		ssun1 3367	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
5		sseq2 3248	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐴 ⊆ ∅))
6	4, 5	mpbii 148	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 ⊆ ∅)
7		ss0b 3531	. . . 4 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
8	6, 7	sylib 122	. . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 = ∅)
9		ssun2 3368	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
10		sseq2 3248	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐵 ⊆ ∅))
11	9, 10	mpbii 148	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 ⊆ ∅)
12		ss0b 3531	. . . 4 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
13	11, 12	sylib 122	. . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 = ∅)
14	8, 13	jca 306	. 2 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))
15	3, 14	impbii 126	1 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395   ∪ cun 3195   ⊆ wss 3197  ∅c0 3491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492

This theorem is referenced by:  undisj1  3549  undisj2  3550  disjpr2  3730