Theorem un00 1728

Description: Two classes are empty iff their union is empty.

Assertion

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

Proof of Theorem un00

Step	Hyp	Ref	Expression
1		uneq12 1613	. . 3 ⊢ ((A = ∅ ∧ B = ∅) → (A ∪ B) = (∅ ∪ ∅))
2		un0 1721	. . 3 ⊢ (∅ ∪ ∅) = ∅
3	1, 2	syl6eq 1140	. 2 ⊢ ((A = ∅ ∧ B = ∅) → (A ∪ B) = ∅)
4		ssun1 1621	. . . . 5 ⊢ A ⊆ (A ∪ B)
5		sseq2 1522	. . . . 5 ⊢ ((A ∪ B) = ∅ → (A ⊆ (A ∪ B) ↔ A ⊆ ∅))
6	4, 5	mpbii 168	. . . 4 ⊢ ((A ∪ B) = ∅ → A ⊆ ∅)
7		ss0b 1726	. . . 4 ⊢ (A ⊆ ∅ ↔ A = ∅)
8	6, 7	sylib 173	. . 3 ⊢ ((A ∪ B) = ∅ → A = ∅)
9		ssun2 1622	. . . . 5 ⊢ B ⊆ (A ∪ B)
10		sseq2 1522	. . . . 5 ⊢ ((A ∪ B) = ∅ → (B ⊆ (A ∪ B) ↔ B ⊆ ∅))
11	9, 10	mpbii 168	. . . 4 ⊢ ((A ∪ B) = ∅ → B ⊆ ∅)
12		ss0b 1726	. . . 4 ⊢ (B ⊆ ∅ ↔ B = ∅)
13	11, 12	sylib 173	. . 3 ⊢ ((A ∪ B) = ∅ → B = ∅)
14	8, 13	jca 236	. 2 ⊢ ((A ∪ B) = ∅ → (A = ∅ ∧ B = ∅))
15	3, 14	impbi 139	1 ⊢ ((A = ∅ ∧ B = ∅) ↔ (A ∪ B) = ∅)

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∪ cun 1485   ⊆ wss 1487  ∅c0 1707

This theorem is referenced by:  undisj1 1739  undisj2 1740

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708

metamath.org