Theorem elpw12 4145

Description: Membership in a unit power class applied twice. (Contributed by SF, 15-Jan-2015.)

Assertion

Ref	Expression
elpw12	|- (A e. ~P1 ~P1 B <-> E.x e. B A = {{x}})

Distinct variable groups:   x,A   x,B

Proof of Theorem elpw12

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw1 4144	. 2 |- (A e. ~P1 ~P1 B <-> E.y e. ~P1 BA = {y})
2		elpw1 4144	. . . . . 6 |- (y e. ~P1 B <-> E.x e. B y = {x})
3	2	anbi1i 676	. . . . 5 |- ((y e. ~P1 B /\ A = {y}) <-> (E.x e. B y = {x} /\ A = {y}))
4		r19.41v 2764	. . . . 5 |- (E.x e. B (y = {x} /\ A = {y}) <-> (E.x e. B y = {x} /\ A = {y}))
5	3, 4	bitr4i 243	. . . 4 |- ((y e. ~P1 B /\ A = {y}) <-> E.x e. B (y = {x} /\ A = {y}))
6	5	exbii 1582	. . 3 |- (E.y(y e. ~P1 B /\ A = {y}) <-> E.yE.x e. B (y = {x} /\ A = {y}))
7		df-rex 2620	. . 3 |- (E.y e. ~P1 BA = {y} <-> E.y(y e. ~P1 B /\ A = {y}))
8		rexcom4 2878	. . 3 |- (E.x e. B E.y(y = {x} /\ A = {y}) <-> E.yE.x e. B (y = {x} /\ A = {y}))
9	6, 7, 8	3bitr4i 268	. 2 |- (E.y e. ~P1 BA = {y} <-> E.x e. B E.y(y = {x} /\ A = {y}))
10		snex 4111	. . . 4 |- {x} e. _V
11		sneq 3744	. . . . 5 |- (y = {x} -> {y} = {{x}})
12	11	eqeq2d 2364	. . . 4 |- (y = {x} -> (A = {y} <-> A = {{x}}))
13	10, 12	ceqsexv 2894	. . 3 |- (E.y(y = {x} /\ A = {y}) <-> A = {{x}})
14	13	rexbii 2639	. 2 |- (E.x e. B E.y(y = {x} /\ A = {y}) <-> E.x e. B A = {{x}})
15	1, 9, 14	3bitri 262	1 |- (A e. ~P1 ~P1 B <-> E.x e. B A = {{x}})

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2615  {csn 3737  ~P₁ cpw1 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:  dfaddc2  4381  ltfinex  4464  ncfinlowerlem1  4482  nnpweqlem1  4522  setconslem6  4736