Theorem unissint 4996

Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 5009). (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 482	. . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 ⊆ ∩ 𝐴)
2		df-ne 2947	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3		intssuni 4994	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
4	2, 3	sylbir 235	. . . . . 6 ⊢ (¬ 𝐴 = ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
5	4	adantl 481	. . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∩ 𝐴 ⊆ ∪ 𝐴)
6	1, 5	eqssd 4026	. . . 4 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 = ∩ 𝐴)
7	6	ex 412	. . 3 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (¬ 𝐴 = ∅ → ∪ 𝐴 = ∩ 𝐴))
8	7	orrd 862	. 2 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))
9		ssv 4033	. . . . 5 ⊢ ∪ 𝐴 ⊆ V
10		int0 4986	. . . . 5 ⊢ ∩ ∅ = V
11	9, 10	sseqtrri 4046	. . . 4 ⊢ ∪ 𝐴 ⊆ ∩ ∅
12		inteq 4973	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
13	11, 12	sseqtrrid 4062	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ⊆ ∩ 𝐴)
14		eqimss 4067	. . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∪ 𝐴 ⊆ ∩ 𝐴)
15	13, 14	jaoi 856	. 2 ⊢ ((𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴) → ∪ 𝐴 ⊆ ∩ 𝐴)
16	8, 15	impbii 209	1 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ≠ wne 2946  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  ∩ cint 4970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-v 3490  df-dif 3979  df-ss 3993  df-nul 4353  df-uni 4932  df-int 4971

This theorem is referenced by: (None)