Theorem uniintsn 3963

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 3796. (Contributed by NM, 2-Aug-2010.)

Assertion

Ref	Expression
uniintsn	|- (U.A = |^|A <-> E.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 3557	. . . . . 6 |- _V =/= (/)
2		inteq 3929	. . . . . . . . . . 11 |- (A = (/) -> |^|A = |^|(/))
3		int0 3940	. . . . . . . . . . 11 |- |^|(/) = _V
4	2, 3	syl6eq 2401	. . . . . . . . . 10 |- (A = (/) -> |^|A = _V)
5	4	adantl 452	. . . . . . . . 9 |- ((U.A = |^|A /\ A = (/)) -> |^|A = _V)
6		unieq 3900	. . . . . . . . . . . 12 |- (A = (/) -> U.A = U.(/))
7		uni0 3918	. . . . . . . . . . . 12 |- U.(/) = (/)
8	6, 7	syl6eq 2401	. . . . . . . . . . 11 |- (A = (/) -> U.A = (/))
9		eqeq1 2359	. . . . . . . . . . 11 |- (U.A = |^|A -> (U.A = (/) <-> |^|A = (/)))
10	8, 9	syl5ib 210	. . . . . . . . . 10 |- (U.A = |^|A -> (A = (/) -> |^|A = (/)))
11	10	imp 418	. . . . . . . . 9 |- ((U.A = |^|A /\ A = (/)) -> |^|A = (/))
12	5, 11	eqtr3d 2387	. . . . . . . 8 |- ((U.A = |^|A /\ A = (/)) -> _V = (/))
13	12	ex 423	. . . . . . 7 |- (U.A = |^|A -> (A = (/) -> _V = (/)))
14	13	necon3d 2554	. . . . . 6 |- (U.A = |^|A -> (_V =/= (/) -> A =/= (/)))
15	1, 14	mpi 16	. . . . 5 |- (U.A = |^|A -> A =/= (/))
16		n0 3559	. . . . 5 |- (A =/= (/) <-> E.x x e. A)
17	15, 16	sylib 188	. . . 4 |- (U.A = |^|A -> E.x x e. A)
18		vex 2862	. . . . . . 7 |- x e. _V
19		vex 2862	. . . . . . 7 |- y e. _V
20	18, 19	prss 3861	. . . . . 6 |- ((x e. A /\ y e. A) <-> {x, y} C_ A)
21		uniss 3912	. . . . . . . . . . . . 13 |- ({x, y} C_ A -> U.{x, y} C_ U.A)
22	21	adantl 452	. . . . . . . . . . . 12 |- ((U.A = |^|A /\ {x, y} C_ A) -> U.{x, y} C_ U.A)
23		simpl 443	. . . . . . . . . . . 12 |- ((U.A = |^|A /\ {x, y} C_ A) -> U.A = |^|A)
24	22, 23	sseqtrd 3307	. . . . . . . . . . 11 |- ((U.A = |^|A /\ {x, y} C_ A) -> U.{x, y} C_ |^|A)
25		intss 3947	. . . . . . . . . . . 12 |- ({x, y} C_ A -> |^|A C_ |^|{x, y})
26	25	adantl 452	. . . . . . . . . . 11 |- ((U.A = |^|A /\ {x, y} C_ A) -> |^|A C_ |^|{x, y})
27	24, 26	sstrd 3282	. . . . . . . . . 10 |- ((U.A = |^|A /\ {x, y} C_ A) -> U.{x, y} C_ |^|{x, y})
28	18, 19	unipr 3905	. . . . . . . . . 10 |- U.{x, y} = (x u. y)
29	18, 19	intpr 3959	. . . . . . . . . 10 |- |^|{x, y} = (x i^i y)
30	27, 28, 29	3sstr3g 3311	. . . . . . . . 9 |- ((U.A = |^|A /\ {x, y} C_ A) -> (x u. y) C_ (x i^i y))
31		inss1 3475	. . . . . . . . . 10 |- (x i^i y) C_ x
32		ssun1 3426	. . . . . . . . . 10 |- x C_ (x u. y)
33	31, 32	sstri 3281	. . . . . . . . 9 |- (x i^i y) C_ (x u. y)
34	30, 33	jctir 524	. . . . . . . 8 |- ((U.A = |^|A /\ {x, y} C_ A) -> ((x u. y) C_ (x i^i y) /\ (x i^i y) C_ (x u. y)))
35		eqss 3287	. . . . . . . . 9 |- ((x u. y) = (x i^i y) <-> ((x u. y) C_ (x i^i y) /\ (x i^i y) C_ (x u. y)))
36		uneqin 3506	. . . . . . . . 9 |- ((x u. y) = (x i^i y) <-> x = y)
37	35, 36	bitr3i 242	. . . . . . . 8 |- (((x u. y) C_ (x i^i y) /\ (x i^i y) C_ (x u. y)) <-> x = y)
38	34, 37	sylib 188	. . . . . . 7 |- ((U.A = |^|A /\ {x, y} C_ A) -> x = y)
39	38	ex 423	. . . . . 6 |- (U.A = |^|A -> ({x, y} C_ A -> x = y))
40	20, 39	syl5bi 208	. . . . 5 |- (U.A = |^|A -> ((x e. A /\ y e. A) -> x = y))
41	40	alrimivv 1632	. . . 4 |- (U.A = |^|A -> A.xA.y((x e. A /\ y e. A) -> x = y))
42	17, 41	jca 518	. . 3 |- (U.A = |^|A -> (E.x x e. A /\ A.xA.y((x e. A /\ y e. A) -> x = y)))
43		euabsn 3792	. . . 4 |- (E!x x e. A <-> E.x{x | x e. A} = {x})
44		eleq1 2413	. . . . 5 |- (x = y -> (x e. A <-> y e. A))
45	44	eu4 2243	. . . 4 |- (E!x x e. A <-> (E.x x e. A /\ A.xA.y((x e. A /\ y e. A) -> x = y)))
46		abid2 2470	. . . . . 6 |- {x | x e. A} = A
47	46	eqeq1i 2360	. . . . 5 |- ({x | x e. A} = {x} <-> A = {x})
48	47	exbii 1582	. . . 4 |- (E.x{x | x e. A} = {x} <-> E.x A = {x})
49	43, 45, 48	3bitr3i 266	. . 3 |- ((E.x x e. A /\ A.xA.y((x e. A /\ y e. A) -> x = y)) <-> E.x A = {x})
50	42, 49	sylib 188	. 2 |- (U.A = |^|A -> E.x A = {x})
51	18	unisn 3907	. . . 4 |- U.{x} = x
52		unieq 3900	. . . 4 |- (A = {x} -> U.A = U.{x})
53		inteq 3929	. . . . 5 |- (A = {x} -> |^|A = |^|{x})
54	18	intsn 3962	. . . . 5 |- |^|{x} = x
55	53, 54	syl6eq 2401	. . . 4 |- (A = {x} -> |^|A = x)
56	51, 52, 55	3eqtr4a 2411	. . 3 |- (A = {x} -> U.A = |^|A)
57	56	exlimiv 1634	. 2 |- (E.x A = {x} -> U.A = |^|A)
58	50, 57	impbii 180	1 |- (U.A = |^|A <-> E.x A = {x})

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  {cab 2339   =/= wne 2516  _Vcvv 2859   u. cun 3207   i^i cin 3208   C_ wss 3257  (/)c0 3550  {csn 3737  {cpr 3738  U.cuni 3891  |^|cint 3926

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-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927

This theorem is referenced by:  uniintab  3964