Theorem sikss1c1c 4268

Description: A Kuratowski singleton image is a subset of (1_c X._k 1_c). (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
sikss1c1c	|- SIk A C_ (1c X.k 1c)

Proof of Theorem sikss1c1c

Dummy variables x y z w t a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sik 4193	. . . . 5 |- SIk A = {t | E.zE.w(t = <<z, w>> /\ E.aE.b(z = {a} /\ w = {b} /\ <<a, b>> e. A))}
2		eqeq1 2359	. . . . . . 7 |- (z = x -> (z = {a} <-> x = {a}))
3	2	3anbi1d 1256	. . . . . 6 |- (z = x -> ((z = {a} /\ w = {b} /\ <<a, b>> e. A) <-> (x = {a} /\ w = {b} /\ <<a, b>> e. A)))
4	3	2exbidv 1628	. . . . 5 |- (z = x -> (E.aE.b(z = {a} /\ w = {b} /\ <<a, b>> e. A) <-> E.aE.b(x = {a} /\ w = {b} /\ <<a, b>> e. A)))
5		eqeq1 2359	. . . . . . 7 |- (w = y -> (w = {b} <-> y = {b}))
6	5	3anbi2d 1257	. . . . . 6 |- (w = y -> ((x = {a} /\ w = {b} /\ <<a, b>> e. A) <-> (x = {a} /\ y = {b} /\ <<a, b>> e. A)))
7	6	2exbidv 1628	. . . . 5 |- (w = y -> (E.aE.b(x = {a} /\ w = {b} /\ <<a, b>> e. A) <-> E.aE.b(x = {a} /\ y = {b} /\ <<a, b>> e. A)))
8		vex 2863	. . . . 5 |- x e. _V
9		vex 2863	. . . . 5 |- y e. _V
10	1, 4, 7, 8, 9	opkelopkab 4247	. . . 4 |- (<<x, y>> e. SIk A <-> E.aE.b(x = {a} /\ y = {b} /\ <<a, b>> e. A))
11		opkeq12 4062	. . . . . . 7 |- ((x = {a} /\ y = {b}) -> <<x, y>> = <<{a}, {b}>>)
12		vex 2863	. . . . . . . . 9 |- a e. _V
13	12	snel1c 4141	. . . . . . . 8 |- {a} e. 1c
14		vex 2863	. . . . . . . . 9 |- b e. _V
15	14	snel1c 4141	. . . . . . . 8 |- {b} e. 1c
16		opkelxpkg 4248	. . . . . . . . 9 |- (({a} e. 1c /\ {b} e. 1c) -> (<<{a}, {b}>> e. (1c X.k 1c) <-> ({a} e. 1c /\ {b} e. 1c)))
17	13, 15, 16	mp2an 653	. . . . . . . 8 |- (<<{a}, {b}>> e. (1c X.k 1c) <-> ({a} e. 1c /\ {b} e. 1c))
18	13, 15, 17	mpbir2an 886	. . . . . . 7 |- <<{a}, {b}>> e. (1c X.k 1c)
19	11, 18	syl6eqel 2441	. . . . . 6 |- ((x = {a} /\ y = {b}) -> <<x, y>> e. (1c X.k 1c))
20	19	3adant3 975	. . . . 5 |- ((x = {a} /\ y = {b} /\ <<a, b>> e. A) -> <<x, y>> e. (1c X.k 1c))
21	20	exlimivv 1635	. . . 4 |- (E.aE.b(x = {a} /\ y = {b} /\ <<a, b>> e. A) -> <<x, y>> e. (1c X.k 1c))
22	10, 21	sylbi 187	. . 3 |- (<<x, y>> e. SIk A -> <<x, y>> e. (1c X.k 1c))
23	22	gen2 1547	. 2 |- A.xA.y(<<x, y>> e. SIk A -> <<x, y>> e. (1c X.k 1c))
24		sikssvvk 4267	. . 3 |- SIk A C_ (_V X.k _V)
25		ssrelk 4212	. . 3 |- ( SIk A C_ (_V X.k _V) -> ( SIk A C_ (1c X.k 1c) <-> A.xA.y(<<x, y>> e. SIk A -> <<x, y>> e. (1c X.k 1c))))
26	24, 25	ax-mp 5	. 2 |- ( SIk A C_ (1c X.k 1c) <-> A.xA.y(<<x, y>> e. SIk A -> <<x, y>> e. (1c X.k 1c)))
27	23, 26	mpbir 200	1 |- SIk A C_ (1c X.k 1c)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860   C_ wss 3258  {csn 3738  <<copk 4058  1_cc1c 4135   X._k cxpk 4175   SI_k csik 4182

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  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-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-opk 4059  df-1c 4137  df-xpk 4186  df-sik 4193

This theorem is referenced by:  opkelimagekg  4272  sikexg  4297  dfnnc2  4396