Theorem uniintsn 3964

Description: Two ways to express "A is a singleton." See also en1 in set.mm, en1b in set.mm, card1 in set.mm, and eusn 3797. (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 3558	. . . . . 6 ⊢ V ≠ ∅
2		inteq 3930	. . . . . . . . . . 11 ⊢ (A = ∅ → ∩A = ∩∅)
3		int0 3941	. . . . . . . . . . 11 ⊢ ∩∅ = V
4	2, 3	syl6eq 2401	. . . . . . . . . 10 ⊢ (A = ∅ → ∩A = V)
5	4	adantl 452	. . . . . . . . 9 ⊢ ((∪A = ∩A ∧ A = ∅) → ∩A = V)
6		unieq 3901	. . . . . . . . . . . 12 ⊢ (A = ∅ → ∪A = ∪∅)
7		uni0 3919	. . . . . . . . . . . 12 ⊢ ∪∅ = ∅
8	6, 7	syl6eq 2401	. . . . . . . . . . 11 ⊢ (A = ∅ → ∪A = ∅)
9		eqeq1 2359	. . . . . . . . . . 11 ⊢ (∪A = ∩A → (∪A = ∅ ↔ ∩A = ∅))
10	8, 9	syl5ib 210	. . . . . . . . . 10 ⊢ (∪A = ∩A → (A = ∅ → ∩A = ∅))
11	10	imp 418	. . . . . . . . 9 ⊢ ((∪A = ∩A ∧ A = ∅) → ∩A = ∅)
12	5, 11	eqtr3d 2387	. . . . . . . 8 ⊢ ((∪A = ∩A ∧ A = ∅) → V = ∅)
13	12	ex 423	. . . . . . 7 ⊢ (∪A = ∩A → (A = ∅ → V = ∅))
14	13	necon3d 2555	. . . . . 6 ⊢ (∪A = ∩A → (V ≠ ∅ → A ≠ ∅))
15	1, 14	mpi 16	. . . . 5 ⊢ (∪A = ∩A → A ≠ ∅)
16		n0 3560	. . . . 5 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A)
17	15, 16	sylib 188	. . . 4 ⊢ (∪A = ∩A → ∃x x ∈ A)
18		vex 2863	. . . . . . 7 ⊢ x ∈ V
19		vex 2863	. . . . . . 7 ⊢ y ∈ V
20	18, 19	prss 3862	. . . . . 6 ⊢ ((x ∈ A ∧ y ∈ A) ↔ {x, y} ⊆ A)
21		uniss 3913	. . . . . . . . . . . . 13 ⊢ ({x, y} ⊆ A → ∪{x, y} ⊆ ∪A)
22	21	adantl 452	. . . . . . . . . . . 12 ⊢ ((∪A = ∩A ∧ {x, y} ⊆ A) → ∪{x, y} ⊆ ∪A)
23		simpl 443	. . . . . . . . . . . 12 ⊢ ((∪A = ∩A ∧ {x, y} ⊆ A) → ∪A = ∩A)
24	22, 23	sseqtrd 3308	. . . . . . . . . . 11 ⊢ ((∪A = ∩A ∧ {x, y} ⊆ A) → ∪{x, y} ⊆ ∩A)
25		intss 3948	. . . . . . . . . . . 12 ⊢ ({x, y} ⊆ A → ∩A ⊆ ∩{x, y})
26	25	adantl 452	. . . . . . . . . . 11 ⊢ ((∪A = ∩A ∧ {x, y} ⊆ A) → ∩A ⊆ ∩{x, y})
27	24, 26	sstrd 3283	. . . . . . . . . 10 ⊢ ((∪A = ∩A ∧ {x, y} ⊆ A) → ∪{x, y} ⊆ ∩{x, y})
28	18, 19	unipr 3906	. . . . . . . . . 10 ⊢ ∪{x, y} = (x ∪ y)
29	18, 19	intpr 3960	. . . . . . . . . 10 ⊢ ∩{x, y} = (x ∩ y)
30	27, 28, 29	3sstr3g 3312	. . . . . . . . 9 ⊢ ((∪A = ∩A ∧ {x, y} ⊆ A) → (x ∪ y) ⊆ (x ∩ y))
31		inss1 3476	. . . . . . . . . 10 ⊢ (x ∩ y) ⊆ x
32		ssun1 3427	. . . . . . . . . 10 ⊢ x ⊆ (x ∪ y)
33	31, 32	sstri 3282	. . . . . . . . 9 ⊢ (x ∩ y) ⊆ (x ∪ y)
34	30, 33	jctir 524	. . . . . . . 8 ⊢ ((∪A = ∩A ∧ {x, y} ⊆ A) → ((x ∪ y) ⊆ (x ∩ y) ∧ (x ∩ y) ⊆ (x ∪ y)))
35		eqss 3288	. . . . . . . . 9 ⊢ ((x ∪ y) = (x ∩ y) ↔ ((x ∪ y) ⊆ (x ∩ y) ∧ (x ∩ y) ⊆ (x ∪ y)))
36		uneqin 3507	. . . . . . . . 9 ⊢ ((x ∪ y) = (x ∩ y) ↔ x = y)
37	35, 36	bitr3i 242	. . . . . . . 8 ⊢ (((x ∪ y) ⊆ (x ∩ y) ∧ (x ∩ y) ⊆ (x ∪ y)) ↔ x = y)
38	34, 37	sylib 188	. . . . . . 7 ⊢ ((∪A = ∩A ∧ {x, y} ⊆ A) → x = y)
39	38	ex 423	. . . . . 6 ⊢ (∪A = ∩A → ({x, y} ⊆ A → x = y))
40	20, 39	syl5bi 208	. . . . 5 ⊢ (∪A = ∩A → ((x ∈ A ∧ y ∈ A) → x = y))
41	40	alrimivv 1632	. . . 4 ⊢ (∪A = ∩A → ∀x∀y((x ∈ A ∧ y ∈ A) → x = y))
42	17, 41	jca 518	. . 3 ⊢ (∪A = ∩A → (∃x x ∈ A ∧ ∀x∀y((x ∈ A ∧ y ∈ A) → x = y)))
43		euabsn 3793	. . . 4 ⊢ (∃!x x ∈ A ↔ ∃x{x ∣ x ∈ A} = {x})
44		eleq1 2413	. . . . 5 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
45	44	eu4 2243	. . . 4 ⊢ (∃!x x ∈ A ↔ (∃x x ∈ A ∧ ∀x∀y((x ∈ A ∧ y ∈ A) → x = y)))
46		abid2 2471	. . . . . 6 ⊢ {x ∣ x ∈ A} = A
47	46	eqeq1i 2360	. . . . 5 ⊢ ({x ∣ x ∈ A} = {x} ↔ A = {x})
48	47	exbii 1582	. . . 4 ⊢ (∃x{x ∣ x ∈ A} = {x} ↔ ∃x A = {x})
49	43, 45, 48	3bitr3i 266	. . 3 ⊢ ((∃x x ∈ A ∧ ∀x∀y((x ∈ A ∧ y ∈ A) → x = y)) ↔ ∃x A = {x})
50	42, 49	sylib 188	. 2 ⊢ (∪A = ∩A → ∃x A = {x})
51	18	unisn 3908	. . . 4 ⊢ ∪{x} = x
52		unieq 3901	. . . 4 ⊢ (A = {x} → ∪A = ∪{x})
53		inteq 3930	. . . . 5 ⊢ (A = {x} → ∩A = ∩{x})
54	18	intsn 3963	. . . . 5 ⊢ ∩{x} = x
55	53, 54	syl6eq 2401	. . . 4 ⊢ (A = {x} → ∩A = x)
56	51, 52, 55	3eqtr4a 2411	. . 3 ⊢ (A = {x} → ∪A = ∩A)
57	56	exlimiv 1634	. 2 ⊢ (∃x A = {x} → ∪A = ∩A)
58	50, 57	impbii 180	1 ⊢ (∪A = ∩A ↔ ∃x A = {x})

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339   ≠ wne 2517  Vcvv 2860   ∪ cun 3208   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551  {csn 3738  {cpr 3739  ∪cuni 3892  ∩cint 3927

This proof depends on 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 proof 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-eu 2208  df-mo 2209  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-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928

This theorem is used by:  uniintab  3965