Theorem unissint 4900

Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4915). (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 2943	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3		intssuni 4898	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
4	2, 3	sylbir 234	. . . . . 6 ⊢ (¬ 𝐴 = ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
5	4	adantl 481	. . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∩ 𝐴 ⊆ ∪ 𝐴)
6	1, 5	eqssd 3934	. . . 4 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 = ∩ 𝐴)
7	6	ex 412	. . 3 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (¬ 𝐴 = ∅ → ∪ 𝐴 = ∩ 𝐴))
8	7	orrd 859	. 2 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))
9		ssv 3941	. . . . 5 ⊢ ∪ 𝐴 ⊆ V
10		int0 4890	. . . . 5 ⊢ ∩ ∅ = V
11	9, 10	sseqtrri 3954	. . . 4 ⊢ ∪ 𝐴 ⊆ ∩ ∅
12		inteq 4879	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
13	11, 12	sseqtrrid 3970	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ⊆ ∩ 𝐴)
14		eqimss 3973	. . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∪ 𝐴 ⊆ ∩ 𝐴)
15	13, 14	jaoi 853	. 2 ⊢ ((𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴) → ∪ 𝐴 ⊆ ∩ 𝐴)
16	8, 15	impbii 208	1 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ≠ wne 2942  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  ∩ cint 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-uni 4837  df-int 4877

This theorem is referenced by: (None)