Theorem elpw121c 4149

Description: Membership in ~P₁ ~P₁ 1_c. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
elpw121c	|- (A e. ~P1 ~P1 1c <-> E.x A = {{{x}}})

Distinct variable group:   x,A

Proof of Theorem elpw121c

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw1 4145	. 2 |- (A e. ~P1 ~P1 1c <-> E.y e. ~P1 1cA = {y})
2		df-rex 2621	. . . 4 |- (E.y e. ~P1 1cA = {y} <-> E.y(y e. ~P1 1c /\ A = {y}))
3		elpw11c 4148	. . . . . . 7 |- (y e. ~P1 1c <-> E.x y = {{x}})
4	3	anbi1i 676	. . . . . 6 |- ((y e. ~P1 1c /\ A = {y}) <-> (E.x y = {{x}} /\ A = {y}))
5		19.41v 1901	. . . . . 6 |- (E.x(y = {{x}} /\ A = {y}) <-> (E.x y = {{x}} /\ A = {y}))
6	4, 5	bitr4i 243	. . . . 5 |- ((y e. ~P1 1c /\ A = {y}) <-> E.x(y = {{x}} /\ A = {y}))
7	6	exbii 1582	. . . 4 |- (E.y(y e. ~P1 1c /\ A = {y}) <-> E.yE.x(y = {{x}} /\ A = {y}))
8	2, 7	bitri 240	. . 3 |- (E.y e. ~P1 1cA = {y} <-> E.yE.x(y = {{x}} /\ A = {y}))
9		excom 1741	. . . 4 |- (E.yE.x(y = {{x}} /\ A = {y}) <-> E.xE.y(y = {{x}} /\ A = {y}))
10		snex 4112	. . . . . 6 |- {{x}} e. _V
11		sneq 3745	. . . . . . 7 |- (y = {{x}} -> {y} = {{{x}}})
12	11	eqeq2d 2364	. . . . . 6 |- (y = {{x}} -> (A = {y} <-> A = {{{x}}}))
13	10, 12	ceqsexv 2895	. . . . 5 |- (E.y(y = {{x}} /\ A = {y}) <-> A = {{{x}}})
14	13	exbii 1582	. . . 4 |- (E.xE.y(y = {{x}} /\ A = {y}) <-> E.x A = {{{x}}})
15	9, 14	bitri 240	. . 3 |- (E.yE.x(y = {{x}} /\ A = {y}) <-> E.x A = {{{x}}})
16	8, 15	bitri 240	. 2 |- (E.y e. ~P1 1cA = {y} <-> E.x A = {{{x}}})
17	1, 16	bitri 240	1 |- (A e. ~P1 ~P1 1c <-> E.x A = {{{x}}})

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616  {csn 3738  1_cc1c 4135  ~P₁ cpw1 4136

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-1c 4137  df-pw1 4138

This theorem is referenced by:  elpw131c  4150  opkelimagekg  4272  ndisjrelk  4324  eqpwrelk  4479  ncfinraiselem2  4481  ncfinlowerlem1  4483  eqtfinrelk  4487  evenfinex  4504  oddfinex  4505  evenodddisjlem1  4516  nnadjoinlem1  4520  nnpweqlem1  4523  srelk  4525  tfinnnlem1  4534  spfinex  4538  dfop2lem1  4574  setconslem2  4733