Theorem uniintsn 4954

Description: Two ways to express "𝐴 is a singleton". See also en1 9021, en1b 9022, card1 9954, and eusn 4701. (Contributed by NM, 2-Aug-2010.)

Assertion

Ref	Expression
uniintsn	⊢ (∪ 𝐴 = ∩ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Distinct variable group:   𝑥,𝐴

Proof of Theorem uniintsn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vn0 4306	. . . . . 6 ⊢ V ≠ ∅
2		inteq 4919	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
3		int0 4931	. . . . . . . . . . 11 ⊢ ∩ ∅ = V
4	2, 3	eqtrdi 2820	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
5	4	adantl 486	. . . . . . . . 9 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ 𝐴 = ∅) → ∩ 𝐴 = V)
6		unieq 4887	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
7		uni0 4905	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
8	6, 7	eqtrdi 2820	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
9		eqeq1 2773	. . . . . . . . . . 11 ⊢ (∪ 𝐴 = ∩ 𝐴 → (∪ 𝐴 = ∅ ↔ ∩ 𝐴 = ∅))
10	8, 9	imbitrid 247	. . . . . . . . . 10 ⊢ (∪ 𝐴 = ∩ 𝐴 → (𝐴 = ∅ → ∩ 𝐴 = ∅))
11	10	imp 411	. . . . . . . . 9 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ 𝐴 = ∅) → ∩ 𝐴 = ∅)
12	5, 11	eqtr3d 2806	. . . . . . . 8 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ 𝐴 = ∅) → V = ∅)
13	12	ex 417	. . . . . . 7 ⊢ (∪ 𝐴 = ∩ 𝐴 → (𝐴 = ∅ → V = ∅))
14	13	necon3d 2985	. . . . . 6 ⊢ (∪ 𝐴 = ∩ 𝐴 → (V ≠ ∅ → 𝐴 ≠ ∅))
15	1, 14	mpi 21	. . . . 5 ⊢ (∪ 𝐴 = ∩ 𝐴 → 𝐴 ≠ ∅)
16		n0 4315	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
17	15, 16	sylib 221	. . . 4 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
18		vex 3467	. . . . . . 7 ⊢ 𝑥 ∈ V
19		vex 3467	. . . . . . 7 ⊢ 𝑦 ∈ V
20	18, 19	prss 4790	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ {𝑥, 𝑦} ⊆ 𝐴)
21		uniss 4884	. . . . . . . . . . . 12 ⊢ ({𝑥, 𝑦} ⊆ 𝐴 → ∪ {𝑥, 𝑦} ⊆ ∪ 𝐴)
22	21	adantl 486	. . . . . . . . . . 11 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ {𝑥, 𝑦} ⊆ ∪ 𝐴)
23		simpl 487	. . . . . . . . . . 11 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ 𝐴 = ∩ 𝐴)
24	22, 23	sseqtrd 3981	. . . . . . . . . 10 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ {𝑥, 𝑦} ⊆ ∩ 𝐴)
25		intss 4938	. . . . . . . . . . 11 ⊢ ({𝑥, 𝑦} ⊆ 𝐴 → ∩ 𝐴 ⊆ ∩ {𝑥, 𝑦})
26	25	adantl 486	. . . . . . . . . 10 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∩ 𝐴 ⊆ ∩ {𝑥, 𝑦})
27	24, 26	sstrd 3955	. . . . . . . . 9 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ {𝑥, 𝑦} ⊆ ∩ {𝑥, 𝑦})
28	18, 19	unipr 4893	. . . . . . . . 9 ⊢ ∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦)
29	18, 19	intpr 4951	. . . . . . . . 9 ⊢ ∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦)
30	27, 28, 29	3sstr3g 3997	. . . . . . . 8 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → (𝑥 ∪ 𝑦) ⊆ (𝑥 ∩ 𝑦))
31		inss1 4197	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
32		ssun1 4139	. . . . . . . . 9 ⊢ 𝑥 ⊆ (𝑥 ∪ 𝑦)
33	31, 32	sstri 3954	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∪ 𝑦)
34		eqss 3960	. . . . . . . . 9 ⊢ ((𝑥 ∪ 𝑦) = (𝑥 ∩ 𝑦) ↔ ((𝑥 ∪ 𝑦) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∪ 𝑦)))
35		uneqin 4250	. . . . . . . . 9 ⊢ ((𝑥 ∪ 𝑦) = (𝑥 ∩ 𝑦) ↔ 𝑥 = 𝑦)
36	34, 35	bitr3i 280	. . . . . . . 8 ⊢ (((𝑥 ∪ 𝑦) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∪ 𝑦)) ↔ 𝑥 = 𝑦)
37	30, 33, 36	sylanblc 600	. . . . . . 7 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝑥 = 𝑦)
38	37	ex 417	. . . . . 6 ⊢ (∪ 𝐴 = ∩ 𝐴 → ({𝑥, 𝑦} ⊆ 𝐴 → 𝑥 = 𝑦))
39	20, 38	biimtrid 245	. . . . 5 ⊢ (∪ 𝐴 = ∩ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
40	39	alrimivv 1955	. . . 4 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
41	17, 40	jca 520	. . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → (∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)))
42		euabsn 4697	. . . 4 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥})
43		eleq1w 2852	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
44	43	eu4 2649	. . . 4 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)))
45		abid2 2906	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
46	45	eqeq1i 2774	. . . . 5 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
47	46	exbii 1875	. . . 4 ⊢ (∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
48	42, 44, 47	3bitr3i 304	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) ↔ ∃𝑥 𝐴 = {𝑥})
49	41, 48	sylib 221	. 2 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∃𝑥 𝐴 = {𝑥})
50		unisnv 4896	. . . 4 ⊢ ∪ {𝑥} = 𝑥
51		unieq 4887	. . . 4 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
52		inteq 4919	. . . . 5 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = ∩ {𝑥})
53	18	intsn 4953	. . . . 5 ⊢ ∩ {𝑥} = 𝑥
54	52, 53	eqtrdi 2820	. . . 4 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = 𝑥)
55	50, 51, 54	3eqtr4a 2830	. . 3 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)
56	55	exlimiv 1957	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)
57	49, 56	impbii 212	1 ⊢ (∪ 𝐴 = ∩ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃!weu 2602  {cab 2747   ≠ wne 2964  Vcvv 3463   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  {csn 4594  {cpr 4596  ∪ cuni 4876  ∩ cint 4916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-sn 4595  df-pr 4597  df-uni 4877  df-int 4917

This theorem is referenced by:  uniintab  4955