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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 SkIns3k SIk Sk ) “k 111c) ↔ B = A)

Proof of Theorem eqpwrelk
Dummy variables x t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opkex 4113 . . . . 5 ⟪{A}, B V
21elimak 4259 . . . 4 (⟪{A}, B (( Ins2k SkIns3k SIk Sk ) “k 111c) ↔ t 1 11ct, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk ))
3 elpw121c 4148 . . . . . . . 8 (t 111cx t = {{{x}}})
43anbi1i 676 . . . . . . 7 ((t 111c t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )) ↔ (x t = {{{x}}} t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )))
5 19.41v 1901 . . . . . . 7 (x(t = {{{x}}} t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )) ↔ (x t = {{{x}}} t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )))
64, 5bitr4i 243 . . . . . 6 ((t 111c t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )) ↔ x(t = {{{x}}} t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )))
76exbii 1582 . . . . 5 (t(t 111c t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )) ↔ tx(t = {{{x}}} t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )))
8 df-rex 2620 . . . . 5 (t 1 11ct, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk ) ↔ t(t 111c t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )))
9 excom 1741 . . . . 5 (xt(t = {{{x}}} t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )) ↔ tx(t = {{{x}}} t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )))
107, 8, 93bitr4i 268 . . . 4 (t 1 11ct, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk ) ↔ xt(t = {{{x}}} t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )))
11 snex 4111 . . . . . . 7 {{{x}}} V
12 opkeq1 4059 . . . . . . . 8 (t = {{{x}}} → ⟪t, ⟪{A}, B⟫⟫ = ⟪{{{x}}}, ⟪{A}, B⟫⟫)
1312eleq1d 2419 . . . . . . 7 (t = {{{x}}} → (⟪t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk ) ↔ ⟪{{{x}}}, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )))
1411, 13ceqsexv 2894 . . . . . 6 (t(t = {{{x}}} t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )) ↔ ⟪{{{x}}}, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk ))
15 elsymdif 3223 . . . . . 6 (⟪{{{x}}}, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k 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
1916, 17, 18otkelins2k 4255 . . . . . . . . 9 (⟪{{{x}}}, ⟪{A}, B⟫⟫ Ins2k Sk ↔ ⟪{x}, B Sk )
20 vex 2862 . . . . . . . . . 10 x V
2120, 18elssetk 4270 . . . . . . . . 9 (⟪{x}, B Skx B)
2219, 21bitri 240 . . . . . . . 8 (⟪{{{x}}}, ⟪{A}, B⟫⟫ Ins2k Skx B)
2316, 17, 18otkelins3k 4256 . . . . . . . . 9 (⟪{{{x}}}, ⟪{A}, B⟫⟫ Ins3k SIk Sk ↔ ⟪{x}, {A}⟫ SIk Sk )
24 eqpwrelk.1 . . . . . . . . . 10 A V
2520, 24opksnelsik 4265 . . . . . . . . 9 (⟪{x}, {A}⟫ SIk Sk ↔ ⟪x, A Sk )
26 opkelssetkg 4268 . . . . . . . . . 10 ((x V A V) → (⟪x, A Skx A))
2720, 24, 26mp2an 653 . . . . . . . . 9 (⟪x, A Skx A)
2823, 25, 273bitri 262 . . . . . . . 8 (⟪{{{x}}}, ⟪{A}, B⟫⟫ Ins3k SIk Skx A)
2922, 28bibi12i 306 . . . . . . 7 ((⟪{{{x}}}, ⟪{A}, B⟫⟫ Ins2k Sk ↔ ⟪{{{x}}}, ⟪{A}, B⟫⟫ Ins3k SIk Sk ) ↔ (x Bx A))
3029notbii 287 . . . . . 6 (¬ (⟪{{{x}}}, ⟪{A}, B⟫⟫ Ins2k Sk ↔ ⟪{{{x}}}, ⟪{A}, B⟫⟫ Ins3k SIk Sk ) ↔ ¬ (x Bx A))
3114, 15, 303bitri 262 . . . . 5 (t(t = {{{x}}} t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )) ↔ ¬ (x Bx A))
3231exbii 1582 . . . 4 (xt(t = {{{x}}} t, ⟪{A}, B⟫⟫ ( Ins2k SkIns3k SIk Sk )) ↔ x ¬ (x Bx A))
332, 10, 323bitri 262 . . 3 (⟪{A}, B (( Ins2k SkIns3k SIk Sk ) “k 111c) ↔ x ¬ (x Bx A))
3433notbii 287 . 2 (¬ ⟪{A}, B (( Ins2k SkIns3k SIk Sk ) “k 111c) ↔ ¬ x ¬ (x Bx A))
351elcompl 3225 . 2 (⟪{A}, B ∼ (( Ins2k SkIns3k SIk Sk ) “k 111c) ↔ ¬ ⟪{A}, B (( Ins2k SkIns3k SIk Sk ) “k 111c))
36 df-pw 3724 . . . 4 A = {x x A}
3736eqeq2i 2363 . . 3 (B = AB = {x x A})
38 abeq2 2458 . . 3 (B = {x x A} ↔ x(x Bx A))
39 alex 1572 . . 3 (x(x Bx A) ↔ ¬ x ¬ (x Bx A))
4037, 38, 393bitri 262 . 2 (B = A ↔ ¬ x ¬ (x Bx A))
4134, 35, 403bitr4i 268 1 (⟪{A}, B ∼ (( Ins2k SkIns3k SIk Sk ) “k 111c) ↔ B = A)
 Colors of variables: wff setvar class 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  1cc1c 4134  ℘1cpw1 4135   Ins2k cins2k 4176   Ins3k cins3k 4177   “k cimak 4179   SIk csik 4181   Sk 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
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