Theorem eqpw1relk 4480

Description: Represent equality to unit power class via a Kuratowski relationship. (Contributed by SF, 21-Jan-2015.)

Hypotheses

Ref	Expression
eqpw1relk.1	⊢ A ∈ V
eqpw1relk.2	⊢ B ∈ V

Assertion

Ref	Expression
eqpw1relk	⊢ (⟪A, {B}⟫ ∈ ((℘1c ×k V) ∖ (( Ins3k Sk ⊕ Ins2k SIk Sk ) “k ℘1℘1℘11c)) ↔ A = ℘1B)

Proof of Theorem eqpw1relk

Dummy variables x t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4112	. . . . 5 ⊢ {B} ∈ V
2		eqpw1relk.1	. . . . . 6 ⊢ A ∈ V
3	2, 1	opkelxpk 4249	. . . . 5 ⊢ (⟪A, {B}⟫ ∈ (℘1c ×k V) ↔ (A ∈ ℘1c ∧ {B} ∈ V))
4	1, 3	mpbiran2 885	. . . 4 ⊢ (⟪A, {B}⟫ ∈ (℘1c ×k V) ↔ A ∈ ℘1c)
5	2	elpw 3729	. . . 4 ⊢ (A ∈ ℘1c ↔ A ⊆ 1c)
6	4, 5	bitri 240	. . 3 ⊢ (⟪A, {B}⟫ ∈ (℘1c ×k V) ↔ A ⊆ 1c)
7		opkex 4114	. . . . . . 7 ⊢ ⟪A, {B}⟫ ∈ V
8	7	elimak 4260	. . . . . 6 ⊢ (⟪A, {B}⟫ ∈ (( Ins3k Sk ⊕ Ins2k SIk Sk ) “k ℘1℘1℘11c) ↔ ∃t ∈ ℘1 ℘1℘11c⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk ))
9		elpw131c 4150	. . . . . . . . . 10 ⊢ (t ∈ ℘1℘1℘11c ↔ ∃x t = {{{{x}}}})
10	9	anbi1i 676	. . . . . . . . 9 ⊢ ((t ∈ ℘1℘1℘11c ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )) ↔ (∃x t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )))
11		19.41v 1901	. . . . . . . . 9 ⊢ (∃x(t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )) ↔ (∃x t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )))
12	10, 11	bitr4i 243	. . . . . . . 8 ⊢ ((t ∈ ℘1℘1℘11c ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )) ↔ ∃x(t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )))
13	12	exbii 1582	. . . . . . 7 ⊢ (∃t(t ∈ ℘1℘1℘11c ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )) ↔ ∃t∃x(t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )))
14		df-rex 2621	. . . . . . 7 ⊢ (∃t ∈ ℘1 ℘1℘11c⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk ) ↔ ∃t(t ∈ ℘1℘1℘11c ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )))
15		excom 1741	. . . . . . 7 ⊢ (∃x∃t(t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )) ↔ ∃t∃x(t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )))
16	13, 14, 15	3bitr4i 268	. . . . . 6 ⊢ (∃t ∈ ℘1 ℘1℘11c⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk ) ↔ ∃x∃t(t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )))
17	8, 16	bitri 240	. . . . 5 ⊢ (⟪A, {B}⟫ ∈ (( Ins3k Sk ⊕ Ins2k SIk Sk ) “k ℘1℘1℘11c) ↔ ∃x∃t(t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )))
18		snex 4112	. . . . . . . 8 ⊢ {{{{x}}}} ∈ V
19		opkeq1 4060	. . . . . . . . 9 ⊢ (t = {{{{x}}}} → ⟪t, ⟪A, {B}⟫⟫ = ⟪{{{{x}}}}, ⟪A, {B}⟫⟫)
20	19	eleq1d 2419	. . . . . . . 8 ⊢ (t = {{{{x}}}} → (⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk ) ↔ ⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )))
21	18, 20	ceqsexv 2895	. . . . . . 7 ⊢ (∃t(t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )) ↔ ⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk ))
22		elsymdif 3224	. . . . . . . 8 ⊢ (⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk ) ↔ ¬ (⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ Ins3k Sk ↔ ⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ Ins2k SIk Sk ))
23		snex 4112	. . . . . . . . . . 11 ⊢ {{x}} ∈ V
24	23, 2, 1	otkelins3k 4257	. . . . . . . . . 10 ⊢ (⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ Ins3k Sk ↔ ⟪{{x}}, A⟫ ∈ Sk )
25		snex 4112	. . . . . . . . . . 11 ⊢ {x} ∈ V
26	25, 2	elssetk 4271	. . . . . . . . . 10 ⊢ (⟪{{x}}, A⟫ ∈ Sk ↔ {x} ∈ A)
27	24, 26	bitri 240	. . . . . . . . 9 ⊢ (⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ Ins3k Sk ↔ {x} ∈ A)
28	23, 2, 1	otkelins2k 4256	. . . . . . . . . 10 ⊢ (⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ Ins2k SIk Sk ↔ ⟪{{x}}, {B}⟫ ∈ SIk Sk )
29		eqpw1relk.2	. . . . . . . . . . . 12 ⊢ B ∈ V
30	25, 29	opksnelsik 4266	. . . . . . . . . . 11 ⊢ (⟪{{x}}, {B}⟫ ∈ SIk Sk ↔ ⟪{x}, B⟫ ∈ Sk )
31		vex 2863	. . . . . . . . . . . 12 ⊢ x ∈ V
32	31, 29	elssetk 4271	. . . . . . . . . . 11 ⊢ (⟪{x}, B⟫ ∈ Sk ↔ x ∈ B)
33	30, 32	bitri 240	. . . . . . . . . 10 ⊢ (⟪{{x}}, {B}⟫ ∈ SIk Sk ↔ x ∈ B)
34	28, 33	bitri 240	. . . . . . . . 9 ⊢ (⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ Ins2k SIk Sk ↔ x ∈ B)
35	27, 34	bibi12i 306	. . . . . . . 8 ⊢ ((⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ Ins3k Sk ↔ ⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ Ins2k SIk Sk ) ↔ ({x} ∈ A ↔ x ∈ B))
36	22, 35	xchbinx 301	. . . . . . 7 ⊢ (⟪{{{{x}}}}, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk ) ↔ ¬ ({x} ∈ A ↔ x ∈ B))
37	21, 36	bitri 240	. . . . . 6 ⊢ (∃t(t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )) ↔ ¬ ({x} ∈ A ↔ x ∈ B))
38	37	exbii 1582	. . . . 5 ⊢ (∃x∃t(t = {{{{x}}}} ∧ ⟪t, ⟪A, {B}⟫⟫ ∈ ( Ins3k Sk ⊕ Ins2k SIk Sk )) ↔ ∃x ¬ ({x} ∈ A ↔ x ∈ B))
39		exnal 1574	. . . . 5 ⊢ (∃x ¬ ({x} ∈ A ↔ x ∈ B) ↔ ¬ ∀x({x} ∈ A ↔ x ∈ B))
40	17, 38, 39	3bitrri 263	. . . 4 ⊢ (¬ ∀x({x} ∈ A ↔ x ∈ B) ↔ ⟪A, {B}⟫ ∈ (( Ins3k Sk ⊕ Ins2k SIk Sk ) “k ℘1℘1℘11c))
41	40	con1bii 321	. . 3 ⊢ (¬ ⟪A, {B}⟫ ∈ (( Ins3k Sk ⊕ Ins2k SIk Sk ) “k ℘1℘1℘11c) ↔ ∀x({x} ∈ A ↔ x ∈ B))
42	6, 41	anbi12i 678	. 2 ⊢ ((⟪A, {B}⟫ ∈ (℘1c ×k V) ∧ ¬ ⟪A, {B}⟫ ∈ (( Ins3k Sk ⊕ Ins2k SIk Sk ) “k ℘1℘1℘11c)) ↔ (A ⊆ 1c ∧ ∀x({x} ∈ A ↔ x ∈ B)))
43		eldif 3222	. 2 ⊢ (⟪A, {B}⟫ ∈ ((℘1c ×k V) ∖ (( Ins3k Sk ⊕ Ins2k SIk Sk ) “k ℘1℘1℘11c)) ↔ (⟪A, {B}⟫ ∈ (℘1c ×k V) ∧ ¬ ⟪A, {B}⟫ ∈ (( Ins3k Sk ⊕ Ins2k SIk Sk ) “k ℘1℘1℘11c)))
44		eqpw1 4163	. 2 ⊢ (A = ℘1B ↔ (A ⊆ 1c ∧ ∀x({x} ∈ A ↔ x ∈ B)))
45	42, 43, 44	3bitr4i 268	1 ⊢ (⟪A, {B}⟫ ∈ ((℘1c ×k V) ∖ (( Ins3k Sk ⊕ Ins2k SIk Sk ) “k ℘1℘1℘11c)) ↔ A = ℘1B)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   ∖ cdif 3207   ⊕ csymdif 3210   ⊆ wss 3258  ℘cpw 3723  {csn 3738  ⟪copk 4058  1_cc1c 4135  ℘₁cpw1 4136   ×_k cxpk 4175   Ins2_k cins2k 4177   Ins3_k cins3k 4178   “_k cimak 4180   SI_k csik 4182   S_k cssetk 4184

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  ax-nin 4079  ax-sn 4088

This proof 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-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-pw1 4138  df-xpk 4186  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-sik 4193  df-ssetk 4194

This theorem is used by:  ncfinraiselem2  4481  ncfinlowerlem1  4483  eqtfinrelk  4487  srelk  4525