Theorem elpw131c 4149

Description: Membership in ℘₁℘₁℘₁1_c. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
elpw131c	⊢ (A ∈ ℘1℘1℘11c ↔ ∃x A = {{{{x}}}})

Distinct variable group:   x,A

Proof of Theorem elpw131c

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw1 4144	. 2 ⊢ (A ∈ ℘1℘1℘11c ↔ ∃y ∈ ℘1 ℘11cA = {y})
2		df-rex 2620	. . . 4 ⊢ (∃y ∈ ℘1 ℘11cA = {y} ↔ ∃y(y ∈ ℘1℘11c ∧ A = {y}))
3		elpw121c 4148	. . . . . . 7 ⊢ (y ∈ ℘1℘11c ↔ ∃x y = {{{x}}})
4	3	anbi1i 676	. . . . . 6 ⊢ ((y ∈ ℘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℘11c ∧ A = {y}) ↔ ∃x(y = {{{x}}} ∧ A = {y}))
7	6	exbii 1582	. . . 4 ⊢ (∃y(y ∈ ℘1℘11c ∧ A = {y}) ↔ ∃y∃x(y = {{{x}}} ∧ A = {y}))
8	2, 7	bitri 240	. . 3 ⊢ (∃y ∈ ℘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 4111	. . . . . 6 ⊢ {{{x}}} ∈ V
11		sneq 3744	. . . . . . 7 ⊢ (y = {{{x}}} → {y} = {{{{x}}}})
12	11	eqeq2d 2364	. . . . . 6 ⊢ (y = {{{x}}} → (A = {y} ↔ A = {{{{x}}}}))
13	10, 12	ceqsexv 2894	. . . . 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 ℘11cA = {y} ↔ ∃x A = {{{{x}}}})
17	1, 16	bitri 240	1 ⊢ (A ∈ ℘1℘1℘11c ↔ ∃x A = {{{{x}}}})

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  {csn 3737  1_cc1c 4134  ℘₁cpw1 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:  elpw141c  4150  nnsucelrlem1  4424  ltfinex  4464  eqpw1relk  4479  ncfinraiselem2  4480  ncfinlowerlem1  4482  eqtfinrelk  4486  oddfinex  4504  nnpweqlem1  4522  srelk  4524