Theorem elpw1101c 4157

Description: Membership in ℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁℘₁1_c. (Contributed by SF, 24-Jan-2015.)

Assertion

Ref	Expression
elpw1101c	⊢ (A ∈ ℘1℘1℘1℘1℘1℘1℘1℘1℘1℘11c ↔ ∃x A = {{{{{{{{{{{x}}}}}}}}}}})

Distinct variable group:   x,A

Proof of Theorem elpw1101c

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw1 4145	. 2 ⊢ (A ∈ ℘1℘1℘1℘1℘1℘1℘1℘1℘1℘11c ↔ ∃y ∈ ℘1 ℘1℘1℘1℘1℘1℘1℘1℘11cA = {y})
2		df-rex 2621	. . . 4 ⊢ (∃y ∈ ℘1 ℘1℘1℘1℘1℘1℘1℘1℘11cA = {y} ↔ ∃y(y ∈ ℘1℘1℘1℘1℘1℘1℘1℘1℘11c ∧ A = {y}))
3		elpw191c 4156	. . . . . . 7 ⊢ (y ∈ ℘1℘1℘1℘1℘1℘1℘1℘1℘11c ↔ ∃x y = {{{{{{{{{{x}}}}}}}}}})
4	3	anbi1i 676	. . . . . 6 ⊢ ((y ∈ ℘1℘1℘1℘1℘1℘1℘1℘1℘11c ∧ A = {y}) ↔ (∃x y = {{{{{{{{{{x}}}}}}}}}} ∧ A = {y}))
5		19.41v 1901	. . . . . 6 ⊢ (∃x(y = {{{{{{{{{{x}}}}}}}}}} ∧ A = {y}) ↔ (∃x y = {{{{{{{{{{x}}}}}}}}}} ∧ A = {y}))
6	4, 5	bitr4i 243	. . . . 5 ⊢ ((y ∈ ℘1℘1℘1℘1℘1℘1℘1℘1℘11c ∧ A = {y}) ↔ ∃x(y = {{{{{{{{{{x}}}}}}}}}} ∧ A = {y}))
7	6	exbii 1582	. . . 4 ⊢ (∃y(y ∈ ℘1℘1℘1℘1℘1℘1℘1℘1℘11c ∧ A = {y}) ↔ ∃y∃x(y = {{{{{{{{{{x}}}}}}}}}} ∧ A = {y}))
8	2, 7	bitri 240	. . 3 ⊢ (∃y ∈ ℘1 ℘1℘1℘1℘1℘1℘1℘1℘11cA = {y} ↔ ∃y∃x(y = {{{{{{{{{{x}}}}}}}}}} ∧ A = {y}))
9		excom 1741	. . . 4 ⊢ (∃y∃x(y = {{{{{{{{{{x}}}}}}}}}} ∧ A = {y}) ↔ ∃x∃y(y = {{{{{{{{{{x}}}}}}}}}} ∧ A = {y}))
10		snex 4112	. . . . . 6 ⊢ {{{{{{{{{{x}}}}}}}}}} ∈ 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 ⊢ (∃y(y = {{{{{{{{{{x}}}}}}}}}} ∧ A = {y}) ↔ A = {{{{{{{{{{{x}}}}}}}}}}})
14	13	exbii 1582	. . . 4 ⊢ (∃x∃y(y = {{{{{{{{{{x}}}}}}}}}} ∧ A = {y}) ↔ ∃x A = {{{{{{{{{{{x}}}}}}}}}}})
15	9, 14	bitri 240	. . 3 ⊢ (∃y∃x(y = {{{{{{{{{{x}}}}}}}}}} ∧ A = {y}) ↔ ∃x A = {{{{{{{{{{{x}}}}}}}}}}})
16	8, 15	bitri 240	. 2 ⊢ (∃y ∈ ℘1 ℘1℘1℘1℘1℘1℘1℘1℘11cA = {y} ↔ ∃x A = {{{{{{{{{{{x}}}}}}}}}}})
17	1, 16	bitri 240	1 ⊢ (A ∈ ℘1℘1℘1℘1℘1℘1℘1℘1℘1℘11c ↔ ∃x A = {{{{{{{{{{{x}}}}}}}}}}})

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  {csn 3738  1_cc1c 4135  ℘₁cpw1 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:  elpw1111c  4158