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Theorem elpw181c 4154
 Description: Membership in ℘1℘1℘1℘1℘1℘1℘1℘11c (Contributed by SF, 15-Jan-2015.)
Assertion
Ref Expression
elpw181c (A 111111111cx A = {{{{{{{{{x}}}}}}}}})
Distinct variable group:   x,A

Proof of Theorem elpw181c
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elpw1 4144 . 2 (A 111111111cy 1 1111111cA = {y})
2 df-rex 2620 . . . 4 (y 1 1111111cA = {y} ↔ y(y 11111111c A = {y}))
3 elpw171c 4153 . . . . . . 7 (y 11111111cx y = {{{{{{{{x}}}}}}}})
43anbi1i 676 . . . . . 6 ((y 11111111c A = {y}) ↔ (x y = {{{{{{{{x}}}}}}}} A = {y}))
5 19.41v 1901 . . . . . 6 (x(y = {{{{{{{{x}}}}}}}} A = {y}) ↔ (x y = {{{{{{{{x}}}}}}}} A = {y}))
64, 5bitr4i 243 . . . . 5 ((y 11111111c A = {y}) ↔ x(y = {{{{{{{{x}}}}}}}} A = {y}))
76exbii 1582 . . . 4 (y(y 11111111c A = {y}) ↔ yx(y = {{{{{{{{x}}}}}}}} A = {y}))
82, 7bitri 240 . . 3 (y 1 1111111cA = {y} ↔ yx(y = {{{{{{{{x}}}}}}}} A = {y}))
9 excom 1741 . . . 4 (yx(y = {{{{{{{{x}}}}}}}} A = {y}) ↔ xy(y = {{{{{{{{x}}}}}}}} A = {y}))
10 snex 4111 . . . . . 6 {{{{{{{{x}}}}}}}} V
11 sneq 3744 . . . . . . 7 (y = {{{{{{{{x}}}}}}}} → {y} = {{{{{{{{{x}}}}}}}}})
1211eqeq2d 2364 . . . . . 6 (y = {{{{{{{{x}}}}}}}} → (A = {y} ↔ A = {{{{{{{{{x}}}}}}}}}))
1310, 12ceqsexv 2894 . . . . 5 (y(y = {{{{{{{{x}}}}}}}} A = {y}) ↔ A = {{{{{{{{{x}}}}}}}}})
1413exbii 1582 . . . 4 (xy(y = {{{{{{{{x}}}}}}}} A = {y}) ↔ x A = {{{{{{{{{x}}}}}}}}})
159, 14bitri 240 . . 3 (yx(y = {{{{{{{{x}}}}}}}} A = {y}) ↔ x A = {{{{{{{{{x}}}}}}}}})
168, 15bitri 240 . 2 (y 1 1111111cA = {y} ↔ x A = {{{{{{{{{x}}}}}}}}})
171, 16bitri 240 1 (A 111111111cx A = {{{{{{{{{x}}}}}}}}})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  {csn 3737  1cc1c 4134  ℘1cpw1 4135 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-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-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-1c 4136  df-pw1 4137 This theorem is referenced by:  elpw191c  4155  ltfinex  4464
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