Theorem unissint 4893

Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4906). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
unissint	⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))

Proof of Theorem unissint

Step	Hyp	Ref	Expression
1		simpl 485	. . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 ⊆ ∩ 𝐴)
2		df-ne 3017	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3		intssuni 4891	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
4	2, 3	sylbir 237	. . . . . 6 ⊢ (¬ 𝐴 = ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
5	4	adantl 484	. . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∩ 𝐴 ⊆ ∪ 𝐴)
6	1, 5	eqssd 3984	. . . 4 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 = ∩ 𝐴)
7	6	ex 415	. . 3 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (¬ 𝐴 = ∅ → ∪ 𝐴 = ∩ 𝐴))
8	7	orrd 859	. 2 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))
9		ssv 3991	. . . . 5 ⊢ ∪ 𝐴 ⊆ V
10		int0 4883	. . . . 5 ⊢ ∩ ∅ = V
11	9, 10	sseqtrri 4004	. . . 4 ⊢ ∪ 𝐴 ⊆ ∩ ∅
12		inteq 4872	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
13	11, 12	sseqtrrid 4020	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ⊆ ∩ 𝐴)
14		eqimss 4023	. . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∪ 𝐴 ⊆ ∩ 𝐴)
15	13, 14	jaoi 853	. 2 ⊢ ((𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴) → ∪ 𝐴 ⊆ ∩ 𝐴)
16	8, 15	impbii 211	1 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ≠ wne 3016  Vcvv 3495   ⊆ wss 3936  ∅c0 4291  ∪ cuni 4832  ∩ cint 4869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-v 3497  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-uni 4833  df-int 4870

This theorem is referenced by: (None)