Theorem unissint 4929

Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4942). (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 486	. . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 ⊆ ∩ 𝐴)
2		df-ne 2957	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3		intssuni 4927	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
4	2, 3	sylbir 237	. . . . . 6 ⊢ (¬ 𝐴 = ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
5	4	adantl 485	. . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∩ 𝐴 ⊆ ∪ 𝐴)
6	1, 5	eqssd 3953	. . . 4 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 = ∩ 𝐴)
7	6	ex 416	. . 3 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (¬ 𝐴 = ∅ → ∪ 𝐴 = ∩ 𝐴))
8	7	orrd 874	. 2 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))
9		ssv 3960	. . . . 5 ⊢ ∪ 𝐴 ⊆ V
10		int0 4919	. . . . 5 ⊢ ∩ ∅ = V
11	9, 10	sseqtrri 3985	. . . 4 ⊢ ∪ 𝐴 ⊆ ∩ ∅
12		inteq 4907	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
13	11, 12	sseqtrrid 3979	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ⊆ ∩ 𝐴)
14		eqimss 3994	. . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∪ 𝐴 ⊆ ∩ 𝐴)
15	13, 14	jaoi 868	. 2 ⊢ ((𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴) → ∪ 𝐴 ⊆ ∩ 𝐴)
16	8, 15	impbii 211	1 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1559   ≠ wne 2956  Vcvv 3453   ⊆ wss 3904  ∅c0 4285  ∪ cuni 4864  ∩ cint 4904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-v 3455  df-dif 3907  df-ss 3921  df-nul 4286  df-uni 4865  df-int 4905

This theorem is referenced by: (None)