Theorem uniintsn 4992

Description: Two ways to express "𝐴 is a singleton". See also en1 9021, en1b 9023, card1 9963, and eusn 4735. (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 4339	. . . . . 6 ⊢ V ≠ ∅
2		inteq 4954	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅)
3		int0 4967	. . . . . . . . . . 11 ⊢ ∩ ∅ = V
4	2, 3	eqtrdi 2789	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V)
5	4	adantl 483	. . . . . . . . 9 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ 𝐴 = ∅) → ∩ 𝐴 = V)
6		unieq 4920	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
7		uni0 4940	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
8	6, 7	eqtrdi 2789	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
9		eqeq1 2737	. . . . . . . . . . 11 ⊢ (∪ 𝐴 = ∩ 𝐴 → (∪ 𝐴 = ∅ ↔ ∩ 𝐴 = ∅))
10	8, 9	imbitrid 243	. . . . . . . . . 10 ⊢ (∪ 𝐴 = ∩ 𝐴 → (𝐴 = ∅ → ∩ 𝐴 = ∅))
11	10	imp 408	. . . . . . . . 9 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ 𝐴 = ∅) → ∩ 𝐴 = ∅)
12	5, 11	eqtr3d 2775	. . . . . . . 8 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ 𝐴 = ∅) → V = ∅)
13	12	ex 414	. . . . . . 7 ⊢ (∪ 𝐴 = ∩ 𝐴 → (𝐴 = ∅ → V = ∅))
14	13	necon3d 2962	. . . . . 6 ⊢ (∪ 𝐴 = ∩ 𝐴 → (V ≠ ∅ → 𝐴 ≠ ∅))
15	1, 14	mpi 20	. . . . 5 ⊢ (∪ 𝐴 = ∩ 𝐴 → 𝐴 ≠ ∅)
16		n0 4347	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
17	15, 16	sylib 217	. . . 4 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
18		vex 3479	. . . . . . 7 ⊢ 𝑥 ∈ V
19		vex 3479	. . . . . . 7 ⊢ 𝑦 ∈ V
20	18, 19	prss 4824	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ {𝑥, 𝑦} ⊆ 𝐴)
21		uniss 4917	. . . . . . . . . . . 12 ⊢ ({𝑥, 𝑦} ⊆ 𝐴 → ∪ {𝑥, 𝑦} ⊆ ∪ 𝐴)
22	21	adantl 483	. . . . . . . . . . 11 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ {𝑥, 𝑦} ⊆ ∪ 𝐴)
23		simpl 484	. . . . . . . . . . 11 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ 𝐴 = ∩ 𝐴)
24	22, 23	sseqtrd 4023	. . . . . . . . . 10 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ {𝑥, 𝑦} ⊆ ∩ 𝐴)
25		intss 4974	. . . . . . . . . . 11 ⊢ ({𝑥, 𝑦} ⊆ 𝐴 → ∩ 𝐴 ⊆ ∩ {𝑥, 𝑦})
26	25	adantl 483	. . . . . . . . . 10 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∩ 𝐴 ⊆ ∩ {𝑥, 𝑦})
27	24, 26	sstrd 3993	. . . . . . . . 9 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ∪ {𝑥, 𝑦} ⊆ ∩ {𝑥, 𝑦})
28	18, 19	unipr 4927	. . . . . . . . 9 ⊢ ∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦)
29	18, 19	intpr 4987	. . . . . . . . 9 ⊢ ∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦)
30	27, 28, 29	3sstr3g 4027	. . . . . . . 8 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → (𝑥 ∪ 𝑦) ⊆ (𝑥 ∩ 𝑦))
31		inss1 4229	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
32		ssun1 4173	. . . . . . . . 9 ⊢ 𝑥 ⊆ (𝑥 ∪ 𝑦)
33	31, 32	sstri 3992	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∪ 𝑦)
34		eqss 3998	. . . . . . . . 9 ⊢ ((𝑥 ∪ 𝑦) = (𝑥 ∩ 𝑦) ↔ ((𝑥 ∪ 𝑦) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∪ 𝑦)))
35		uneqin 4279	. . . . . . . . 9 ⊢ ((𝑥 ∪ 𝑦) = (𝑥 ∩ 𝑦) ↔ 𝑥 = 𝑦)
36	34, 35	bitr3i 277	. . . . . . . 8 ⊢ (((𝑥 ∪ 𝑦) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∪ 𝑦)) ↔ 𝑥 = 𝑦)
37	30, 33, 36	sylanblc 590	. . . . . . 7 ⊢ ((∪ 𝐴 = ∩ 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝑥 = 𝑦)
38	37	ex 414	. . . . . 6 ⊢ (∪ 𝐴 = ∩ 𝐴 → ({𝑥, 𝑦} ⊆ 𝐴 → 𝑥 = 𝑦))
39	20, 38	biimtrid 241	. . . . 5 ⊢ (∪ 𝐴 = ∩ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
40	39	alrimivv 1932	. . . 4 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))
41	17, 40	jca 513	. . 3 ⊢ (∪ 𝐴 = ∩ 𝐴 → (∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)))
42		euabsn 4731	. . . 4 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥})
43		eleq1w 2817	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
44	43	eu4 2612	. . . 4 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)))
45		abid2 2872	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
46	45	eqeq1i 2738	. . . . 5 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
47	46	exbii 1851	. . . 4 ⊢ (∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
48	42, 44, 47	3bitr3i 301	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) ↔ ∃𝑥 𝐴 = {𝑥})
49	41, 48	sylib 217	. 2 ⊢ (∪ 𝐴 = ∩ 𝐴 → ∃𝑥 𝐴 = {𝑥})
50		unisnv 4932	. . . 4 ⊢ ∪ {𝑥} = 𝑥
51		unieq 4920	. . . 4 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
52		inteq 4954	. . . . 5 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = ∩ {𝑥})
53	18	intsn 4991	. . . . 5 ⊢ ∩ {𝑥} = 𝑥
54	52, 53	eqtrdi 2789	. . . 4 ⊢ (𝐴 = {𝑥} → ∩ 𝐴 = 𝑥)
55	50, 51, 54	3eqtr4a 2799	. . 3 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)
56	55	exlimiv 1934	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → ∪ 𝐴 = ∩ 𝐴)
57	49, 56	impbii 208	1 ⊢ (∪ 𝐴 = ∩ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1540   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∃!weu 2563  {cab 2710   ≠ wne 2941  Vcvv 3475   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  {csn 4629  {cpr 4631  ∪ cuni 4909  ∩ cint 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-pr 4632  df-uni 4910  df-int 4952

This theorem is referenced by:  uniintab  4993