Theorem unissint 4938

Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4951). (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 487	. . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 ⊆ ∩ 𝐴)
2		df-ne 2959	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3		intssuni 4936	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
4	2, 3	sylbir 238	. . . . . 6 ⊢ (¬ 𝐴 = ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
5	4	adantl 486	. . . . 5 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∩ 𝐴 ⊆ ∪ 𝐴)
6	1, 5	eqssd 3955	. . . 4 ⊢ ((∪ 𝐴 ⊆ ∩ 𝐴 ∧ ¬ 𝐴 = ∅) → ∪ 𝐴 = ∩ 𝐴)
7	6	ex 417	. . 3 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (¬ 𝐴 = ∅ → ∪ 𝐴 = ∩ 𝐴))
8	7	orrd 876	. 2 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 → (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))
9		ssv 3962	. . . . 5 ⊢ ∪ 𝐴 ⊆ V
10		int0 4928	. . . . 5 ⊢ ∩ ∅ = V
11	9, 10	sseqtrri 3987	. . . 4 ⊢ ∪ 𝐴 ⊆ ∩ ∅
12		inteq 4916	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
13	11, 12	sseqtrrid 3981	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝐴 ⊆ ∩ 𝐴)
14		eqimss 3996	. . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∪ 𝐴 ⊆ ∩ 𝐴)
15	13, 14	jaoi 870	. 2 ⊢ ((𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴) → ∪ 𝐴 ⊆ ∩ 𝐴)
16	8, 15	impbii 212	1 ⊢ (∪ 𝐴 ⊆ ∩ 𝐴 ↔ (𝐴 = ∅ ∨ ∪ 𝐴 = ∩ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ≠ wne 2958  Vcvv 3455   ⊆ wss 3906  ∅c0 4287  ∪ cuni 4873  ∩ cint 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-v 3457  df-dif 3909  df-ss 3923  df-nul 4288  df-uni 4874  df-int 4914

This theorem is referenced by: (None)