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Theorem elpw12 4145
 Description: Membership in a unit power class applied twice. (Contributed by SF, 15-Jan-2015.)
Assertion
Ref Expression
elpw12 (A 11Bx B A = {{x}})
Distinct variable groups:   x,A   x,B

Proof of Theorem elpw12
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elpw1 4144 . 2 (A 11By 1 BA = {y})
2 elpw1 4144 . . . . . 6 (y 1Bx B y = {x})
32anbi1i 676 . . . . 5 ((y 1B A = {y}) ↔ (x B y = {x} A = {y}))
4 r19.41v 2764 . . . . 5 (x B (y = {x} A = {y}) ↔ (x B y = {x} A = {y}))
53, 4bitr4i 243 . . . 4 ((y 1B A = {y}) ↔ x B (y = {x} A = {y}))
65exbii 1582 . . 3 (y(y 1B A = {y}) ↔ yx B (y = {x} A = {y}))
7 df-rex 2620 . . 3 (y 1 BA = {y} ↔ y(y 1B A = {y}))
8 rexcom4 2878 . . 3 (x B y(y = {x} A = {y}) ↔ yx B (y = {x} A = {y}))
96, 7, 83bitr4i 268 . 2 (y 1 BA = {y} ↔ x B y(y = {x} A = {y}))
10 snex 4111 . . . 4 {x} V
11 sneq 3744 . . . . 5 (y = {x} → {y} = {{x}})
1211eqeq2d 2364 . . . 4 (y = {x} → (A = {y} ↔ A = {{x}}))
1310, 12ceqsexv 2894 . . 3 (y(y = {x} A = {y}) ↔ A = {{x}})
1413rexbii 2639 . 2 (x B y(y = {x} A = {y}) ↔ x B A = {{x}})
151, 9, 143bitri 262 1 (A 11Bx B A = {{x}})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  {csn 3737  ℘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:  dfaddc2  4381  ltfinex  4464  ncfinlowerlem1  4482  nnpweqlem1  4522  setconslem6  4736
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