Theorem un00 3290

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 3121	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅))
2		un0 3278	. . 3 ⊢ (∅ ∪ ∅) = ∅
3	1, 2	syl6eq 2129	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ∪ 𝐵) = ∅)
4		ssun1 3135	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
5		sseq2 3021	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐴 ⊆ ∅))
6	4, 5	mpbii 146	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 ⊆ ∅)
7		ss0b 3283	. . . 4 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
8	6, 7	sylib 120	. . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐴 = ∅)
9		ssun2 3136	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
10		sseq2 3021	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∪ 𝐵) ↔ 𝐵 ⊆ ∅))
11	9, 10	mpbii 146	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 ⊆ ∅)
12		ss0b 3283	. . . 4 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
13	11, 12	sylib 120	. . 3 ⊢ ((𝐴 ∪ 𝐵) = ∅ → 𝐵 = ∅)
14	8, 13	jca 300	. 2 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))
15	3, 14	impbii 124	1 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅)

Syntax hints:   ∧ wa 102   ↔ wb 103   = wceq 1284   ∪ cun 2971   ⊆ wss 2973  ∅c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252

This theorem is referenced by:  undisj1  3301  undisj2  3302  disjpr2  3456