Theorem eqpwrelk 4478

Description: Represent equality to power class via a Kuratowski relationship. (Contributed by SF, 26-Jan-2015.)

Hypotheses

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

Assertion

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

Proof of Theorem eqpwrelk

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

Step	Hyp	Ref	Expression
1		opkex 4113	. . . . 5 ⊢ ⟪{A}, B⟫ ∈ V
2	1	elimak 4259	. . . 4 ⊢ (⟪{A}, B⟫ ∈ (( Ins2k Sk ⊕ Ins3k SIk Sk ) “k ℘1℘11c) ↔ ∃t ∈ ℘1 ℘11c⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk ))
3		elpw121c 4148	. . . . . . . 8 ⊢ (t ∈ ℘1℘11c ↔ ∃x t = {{{x}}})
4	3	anbi1i 676	. . . . . . 7 ⊢ ((t ∈ ℘1℘11c ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )) ↔ (∃x t = {{{x}}} ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )))
5		19.41v 1901	. . . . . . 7 ⊢ (∃x(t = {{{x}}} ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )) ↔ (∃x t = {{{x}}} ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )))
6	4, 5	bitr4i 243	. . . . . 6 ⊢ ((t ∈ ℘1℘11c ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )) ↔ ∃x(t = {{{x}}} ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )))
7	6	exbii 1582	. . . . 5 ⊢ (∃t(t ∈ ℘1℘11c ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )) ↔ ∃t∃x(t = {{{x}}} ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )))
8		df-rex 2620	. . . . 5 ⊢ (∃t ∈ ℘1 ℘11c⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk ) ↔ ∃t(t ∈ ℘1℘11c ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )))
9		excom 1741	. . . . 5 ⊢ (∃x∃t(t = {{{x}}} ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )) ↔ ∃t∃x(t = {{{x}}} ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )))
10	7, 8, 9	3bitr4i 268	. . . 4 ⊢ (∃t ∈ ℘1 ℘11c⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk ) ↔ ∃x∃t(t = {{{x}}} ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )))
11		snex 4111	. . . . . . 7 ⊢ {{{x}}} ∈ V
12		opkeq1 4059	. . . . . . . 8 ⊢ (t = {{{x}}} → ⟪t, ⟪{A}, B⟫⟫ = ⟪{{{x}}}, ⟪{A}, B⟫⟫)
13	12	eleq1d 2419	. . . . . . 7 ⊢ (t = {{{x}}} → (⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk ) ↔ ⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )))
14	11, 13	ceqsexv 2894	. . . . . 6 ⊢ (∃t(t = {{{x}}} ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )) ↔ ⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk ))
15		elsymdif 3223	. . . . . 6 ⊢ (⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk ) ↔ ¬ (⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ Ins2k Sk ↔ ⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ Ins3k SIk Sk ))
16		snex 4111	. . . . . . . . . 10 ⊢ {x} ∈ V
17		snex 4111	. . . . . . . . . 10 ⊢ {A} ∈ V
18		eqpwrelk.2	. . . . . . . . . 10 ⊢ B ∈ V
19	16, 17, 18	otkelins2k 4255	. . . . . . . . 9 ⊢ (⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ Ins2k Sk ↔ ⟪{x}, B⟫ ∈ Sk )
20		vex 2862	. . . . . . . . . 10 ⊢ x ∈ V
21	20, 18	elssetk 4270	. . . . . . . . 9 ⊢ (⟪{x}, B⟫ ∈ Sk ↔ x ∈ B)
22	19, 21	bitri 240	. . . . . . . 8 ⊢ (⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ Ins2k Sk ↔ x ∈ B)
23	16, 17, 18	otkelins3k 4256	. . . . . . . . 9 ⊢ (⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ Ins3k SIk Sk ↔ ⟪{x}, {A}⟫ ∈ SIk Sk )
24		eqpwrelk.1	. . . . . . . . . 10 ⊢ A ∈ V
25	20, 24	opksnelsik 4265	. . . . . . . . 9 ⊢ (⟪{x}, {A}⟫ ∈ SIk Sk ↔ ⟪x, A⟫ ∈ Sk )
26		opkelssetkg 4268	. . . . . . . . . 10 ⊢ ((x ∈ V ∧ A ∈ V) → (⟪x, A⟫ ∈ Sk ↔ x ⊆ A))
27	20, 24, 26	mp2an 653	. . . . . . . . 9 ⊢ (⟪x, A⟫ ∈ Sk ↔ x ⊆ A)
28	23, 25, 27	3bitri 262	. . . . . . . 8 ⊢ (⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ Ins3k SIk Sk ↔ x ⊆ A)
29	22, 28	bibi12i 306	. . . . . . 7 ⊢ ((⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ Ins2k Sk ↔ ⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ Ins3k SIk Sk ) ↔ (x ∈ B ↔ x ⊆ A))
30	29	notbii 287	. . . . . 6 ⊢ (¬ (⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ Ins2k Sk ↔ ⟪{{{x}}}, ⟪{A}, B⟫⟫ ∈ Ins3k SIk Sk ) ↔ ¬ (x ∈ B ↔ x ⊆ A))
31	14, 15, 30	3bitri 262	. . . . 5 ⊢ (∃t(t = {{{x}}} ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )) ↔ ¬ (x ∈ B ↔ x ⊆ A))
32	31	exbii 1582	. . . 4 ⊢ (∃x∃t(t = {{{x}}} ∧ ⟪t, ⟪{A}, B⟫⟫ ∈ ( Ins2k Sk ⊕ Ins3k SIk Sk )) ↔ ∃x ¬ (x ∈ B ↔ x ⊆ A))
33	2, 10, 32	3bitri 262	. . 3 ⊢ (⟪{A}, B⟫ ∈ (( Ins2k Sk ⊕ Ins3k SIk Sk ) “k ℘1℘11c) ↔ ∃x ¬ (x ∈ B ↔ x ⊆ A))
34	33	notbii 287	. 2 ⊢ (¬ ⟪{A}, B⟫ ∈ (( Ins2k Sk ⊕ Ins3k SIk Sk ) “k ℘1℘11c) ↔ ¬ ∃x ¬ (x ∈ B ↔ x ⊆ A))
35	1	elcompl 3225	. 2 ⊢ (⟪{A}, B⟫ ∈ ∼ (( Ins2k Sk ⊕ Ins3k SIk Sk ) “k ℘1℘11c) ↔ ¬ ⟪{A}, B⟫ ∈ (( Ins2k Sk ⊕ Ins3k SIk Sk ) “k ℘1℘11c))
36		df-pw 3724	. . . 4 ⊢ ℘A = {x ∣ x ⊆ A}
37	36	eqeq2i 2363	. . 3 ⊢ (B = ℘A ↔ B = {x ∣ x ⊆ A})
38		abeq2 2458	. . 3 ⊢ (B = {x ∣ x ⊆ A} ↔ ∀x(x ∈ B ↔ x ⊆ A))
39		alex 1572	. . 3 ⊢ (∀x(x ∈ B ↔ x ⊆ A) ↔ ¬ ∃x ¬ (x ∈ B ↔ x ⊆ A))
40	37, 38, 39	3bitri 262	. 2 ⊢ (B = ℘A ↔ ¬ ∃x ¬ (x ∈ B ↔ x ⊆ A))
41	34, 35, 40	3bitr4i 268	1 ⊢ (⟪{A}, B⟫ ∈ ∼ (( Ins2k Sk ⊕ Ins3k SIk Sk ) “k ℘1℘11c) ↔ B = ℘A)

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ⊕ csymdif 3209   ⊆ wss 3257  ℘cpw 3722  {csn 3737  ⟪copk 4057  1_cc1c 4134  ℘₁cpw1 4135   Ins2_k cins2k 4176   Ins3_k cins3k 4177   “_k cimak 4179   SI_k csik 4181   S_k cssetk 4183

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 4078  ax-sn 4087

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 2478  df-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-sik 4192  df-ssetk 4193

This theorem is referenced by:  nnpweqlem1  4522