Theorem pw111 4171

Description: The unit power class operation is one-to-one. (Contributed by SF, 26-Feb-2015.)

Assertion

Ref	Expression
pw111	⊢ (℘1A = ℘1B ↔ A = B)

Proof of Theorem pw111

Dummy variables t x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4112	. . . . 5 ⊢ {x} ∈ V
2		eleq1 2413	. . . . . 6 ⊢ (t = {x} → (t ∈ ℘1A ↔ {x} ∈ ℘1A))
3		eleq1 2413	. . . . . 6 ⊢ (t = {x} → (t ∈ ℘1B ↔ {x} ∈ ℘1B))
4	2, 3	bibi12d 312	. . . . 5 ⊢ (t = {x} → ((t ∈ ℘1A ↔ t ∈ ℘1B) ↔ ({x} ∈ ℘1A ↔ {x} ∈ ℘1B)))
5	1, 4	ceqsalv 2886	. . . 4 ⊢ (∀t(t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)) ↔ ({x} ∈ ℘1A ↔ {x} ∈ ℘1B))
6		snelpw1 4147	. . . . 5 ⊢ ({x} ∈ ℘1A ↔ x ∈ A)
7		snelpw1 4147	. . . . 5 ⊢ ({x} ∈ ℘1B ↔ x ∈ B)
8	6, 7	bibi12i 306	. . . 4 ⊢ (({x} ∈ ℘1A ↔ {x} ∈ ℘1B) ↔ (x ∈ A ↔ x ∈ B))
9	5, 8	bitri 240	. . 3 ⊢ (∀t(t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)) ↔ (x ∈ A ↔ x ∈ B))
10	9	albii 1566	. 2 ⊢ (∀x∀t(t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)) ↔ ∀x(x ∈ A ↔ x ∈ B))
11		pw1ss1c 4159	. . . 4 ⊢ ℘1A ⊆ 1c
12		pw1ss1c 4159	. . . 4 ⊢ ℘1B ⊆ 1c
13		ssofeq 4078	. . . 4 ⊢ ((℘1A ⊆ 1c ∧ ℘1B ⊆ 1c) → (℘1A = ℘1B ↔ ∀t ∈ 1c (t ∈ ℘1A ↔ t ∈ ℘1B)))
14	11, 12, 13	mp2an 653	. . 3 ⊢ (℘1A = ℘1B ↔ ∀t ∈ 1c (t ∈ ℘1A ↔ t ∈ ℘1B))
15		df-ral 2620	. . . 4 ⊢ (∀t ∈ 1c (t ∈ ℘1A ↔ t ∈ ℘1B) ↔ ∀t(t ∈ 1c → (t ∈ ℘1A ↔ t ∈ ℘1B)))
16		el1c 4140	. . . . . . . 8 ⊢ (t ∈ 1c ↔ ∃x t = {x})
17	16	imbi1i 315	. . . . . . 7 ⊢ ((t ∈ 1c → (t ∈ ℘1A ↔ t ∈ ℘1B)) ↔ (∃x t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)))
18		19.23v 1891	. . . . . . 7 ⊢ (∀x(t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)) ↔ (∃x t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)))
19	17, 18	bitr4i 243	. . . . . 6 ⊢ ((t ∈ 1c → (t ∈ ℘1A ↔ t ∈ ℘1B)) ↔ ∀x(t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)))
20	19	albii 1566	. . . . 5 ⊢ (∀t(t ∈ 1c → (t ∈ ℘1A ↔ t ∈ ℘1B)) ↔ ∀t∀x(t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)))
21		alcom 1737	. . . . 5 ⊢ (∀t∀x(t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)) ↔ ∀x∀t(t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)))
22	20, 21	bitri 240	. . . 4 ⊢ (∀t(t ∈ 1c → (t ∈ ℘1A ↔ t ∈ ℘1B)) ↔ ∀x∀t(t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)))
23	15, 22	bitri 240	. . 3 ⊢ (∀t ∈ 1c (t ∈ ℘1A ↔ t ∈ ℘1B) ↔ ∀x∀t(t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)))
24	14, 23	bitri 240	. 2 ⊢ (℘1A = ℘1B ↔ ∀x∀t(t = {x} → (t ∈ ℘1A ↔ t ∈ ℘1B)))
25		dfcleq 2347	. 2 ⊢ (A = B ↔ ∀x(x ∈ A ↔ x ∈ B))
26	10, 24, 25	3bitr4i 268	1 ⊢ (℘1A = ℘1B ↔ A = B)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2615   ⊆ wss 3258  {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-ral 2620  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:  pw1fnf1o  5856