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Theorem elpw11c 4147
 Description: Membership in ℘11c (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
elpw11c (A 11cx A = {{x}})
Distinct variable group:   x,A

Proof of Theorem elpw11c
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elpw1 4144 . 2 (A 11cy 1c A = {y})
2 df-rex 2620 . . 3 (y 1c A = {y} ↔ y(y 1c A = {y}))
3 el1c 4139 . . . . . 6 (y 1cx y = {x})
43anbi1i 676 . . . . 5 ((y 1c A = {y}) ↔ (x y = {x} A = {y}))
5 19.41v 1901 . . . . 5 (x(y = {x} A = {y}) ↔ (x y = {x} A = {y}))
64, 5bitr4i 243 . . . 4 ((y 1c A = {y}) ↔ x(y = {x} A = {y}))
76exbii 1582 . . 3 (y(y 1c A = {y}) ↔ yx(y = {x} A = {y}))
82, 7bitri 240 . 2 (y 1c A = {y} ↔ yx(y = {x} A = {y}))
9 excom 1741 . . 3 (yx(y = {x} A = {y}) ↔ xy(y = {x} A = {y}))
10 snex 4111 . . . . 5 {x} V
11 sneq 3744 . . . . . 6 (y = {x} → {y} = {{x}})
1211eqeq2d 2364 . . . . 5 (y = {x} → (A = {y} ↔ A = {{x}}))
1310, 12ceqsexv 2894 . . . 4 (y(y = {x} A = {y}) ↔ A = {{x}})
1413exbii 1582 . . 3 (xy(y = {x} A = {y}) ↔ x A = {{x}})
159, 14bitri 240 . 2 (yx(y = {x} A = {y}) ↔ x A = {{x}})
161, 8, 153bitri 262 1 (A 11cx 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-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-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:  elpw121c  4148  insklem  4304  ncfinraiselem2  4480  ncfinlowerlem1  4482  eqtfinrelk  4486  evenodddisjlem1  4515  nnpweqlem1  4522  sfintfinlem1  4531  tfinnnlem1  4533  df1st2  4738  dfswap2  4741
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