Theorem elpw12 4145

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

Assertion

Ref	Expression
elpw12	⊢ (A ∈ ℘1℘1B ↔ ∃x ∈ 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 ∈ ℘1℘1B ↔ ∃y ∈ ℘1 BA = {y})
2		elpw1 4144	. . . . . 6 ⊢ (y ∈ ℘1B ↔ ∃x ∈ B y = {x})
3	2	anbi1i 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}))
5	3, 4	bitr4i 243	. . . 4 ⊢ ((y ∈ ℘1B ∧ A = {y}) ↔ ∃x ∈ B (y = {x} ∧ A = {y}))
6	5	exbii 1582	. . 3 ⊢ (∃y(y ∈ ℘1B ∧ A = {y}) ↔ ∃y∃x ∈ 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}) ↔ ∃y∃x ∈ B (y = {x} ∧ A = {y}))
9	6, 7, 8	3bitr4i 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}})
12	11	eqeq2d 2364	. . . 4 ⊢ (y = {x} → (A = {y} ↔ A = {{x}}))
13	10, 12	ceqsexv 2894	. . 3 ⊢ (∃y(y = {x} ∧ A = {y}) ↔ A = {{x}})
14	13	rexbii 2639	. 2 ⊢ (∃x ∈ B ∃y(y = {x} ∧ A = {y}) ↔ ∃x ∈ B A = {{x}})
15	1, 9, 14	3bitri 262	1 ⊢ (A ∈ ℘1℘1B ↔ ∃x ∈ B A = {{x}})

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