Theorem unissint 3951

Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 3964). (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 443	. . . . 5 ⊢ ((∪A ⊆ ∩A ∧ ¬ A = ∅) → ∪A ⊆ ∩A)
2		df-ne 2519	. . . . . . 7 ⊢ (A ≠ ∅ ↔ ¬ A = ∅)
3		intssuni 3949	. . . . . . 7 ⊢ (A ≠ ∅ → ∩A ⊆ ∪A)
4	2, 3	sylbir 204	. . . . . 6 ⊢ (¬ A = ∅ → ∩A ⊆ ∪A)
5	4	adantl 452	. . . . 5 ⊢ ((∪A ⊆ ∩A ∧ ¬ A = ∅) → ∩A ⊆ ∪A)
6	1, 5	eqssd 3290	. . . 4 ⊢ ((∪A ⊆ ∩A ∧ ¬ A = ∅) → ∪A = ∩A)
7	6	ex 423	. . 3 ⊢ (∪A ⊆ ∩A → (¬ A = ∅ → ∪A = ∩A))
8	7	orrd 367	. 2 ⊢ (∪A ⊆ ∩A → (A = ∅ ∨ ∪A = ∩A))
9		ssv 3292	. . . . 5 ⊢ ∪A ⊆ V
10		int0 3941	. . . . 5 ⊢ ∩∅ = V
11	9, 10	sseqtr4i 3305	. . . 4 ⊢ ∪A ⊆ ∩∅
12		inteq 3930	. . . 4 ⊢ (A = ∅ → ∩A = ∩∅)
13	11, 12	syl5sseqr 3321	. . 3 ⊢ (A = ∅ → ∪A ⊆ ∩A)
14		eqimss 3324	. . 3 ⊢ (∪A = ∩A → ∪A ⊆ ∩A)
15	13, 14	jaoi 368	. 2 ⊢ ((A = ∅ ∨ ∪A = ∩A) → ∪A ⊆ ∩A)
16	8, 15	impbii 180	1 ⊢ (∪A ⊆ ∩A ↔ (A = ∅ ∨ ∪A = ∩A))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ≠ wne 2517  Vcvv 2860   ⊆ wss 3258  ∅c0 3551  ∪cuni 3892  ∩cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-uni 3893  df-int 3928

This theorem is referenced by: (None)