Theorem un00 3587

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 3414	. . 3 ⊢ ((A = ∅ ∧ B = ∅) → (A ∪ B) = (∅ ∪ ∅))
2		un0 3576	. . 3 ⊢ (∅ ∪ ∅) = ∅
3	1, 2	syl6eq 2401	. 2 ⊢ ((A = ∅ ∧ B = ∅) → (A ∪ B) = ∅)
4		ssun1 3427	. . . . 5 ⊢ A ⊆ (A ∪ B)
5		sseq2 3294	. . . . 5 ⊢ ((A ∪ B) = ∅ → (A ⊆ (A ∪ B) ↔ A ⊆ ∅))
6	4, 5	mpbii 202	. . . 4 ⊢ ((A ∪ B) = ∅ → A ⊆ ∅)
7		ss0b 3581	. . . 4 ⊢ (A ⊆ ∅ ↔ A = ∅)
8	6, 7	sylib 188	. . 3 ⊢ ((A ∪ B) = ∅ → A = ∅)
9		ssun2 3428	. . . . 5 ⊢ B ⊆ (A ∪ B)
10		sseq2 3294	. . . . 5 ⊢ ((A ∪ B) = ∅ → (B ⊆ (A ∪ B) ↔ B ⊆ ∅))
11	9, 10	mpbii 202	. . . 4 ⊢ ((A ∪ B) = ∅ → B ⊆ ∅)
12		ss0b 3581	. . . 4 ⊢ (B ⊆ ∅ ↔ B = ∅)
13	11, 12	sylib 188	. . 3 ⊢ ((A ∪ B) = ∅ → B = ∅)
14	8, 13	jca 518	. 2 ⊢ ((A ∪ B) = ∅ → (A = ∅ ∧ B = ∅))
15	3, 14	impbii 180	1 ⊢ ((A = ∅ ∧ B = ∅) ↔ (A ∪ B) = ∅)

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∪ cun 3208   ⊆ wss 3258  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  undisj1  3603  undisj2  3604  0nelsuc  4401  addcass  4416