Theorem prsspw 1858

Description: An unordered pair belongs to the power class of a class iff each member belongs to the class.

Hypotheses

Ref	Expression
prsspw.1	⊢ A ∈ V
prsspw.2	⊢ B ∈ V

Assertion

Ref	Expression
prsspw	⊢ ({A, B} ⊆ ℘C ↔ (A ⊆ C ∧ B ⊆ C))

Proof of Theorem prsspw

Step	Hyp	Ref	Expression
1		dfss2 1497	. 2 ⊢ ({A, B} ⊆ ℘C ↔ ∀x(x ∈ {A, B} → x ∈ ℘C))
2		visset 1350	. . . . . 6 ⊢ x ∈ V
3	2	elpr 1823	. . . . 5 ⊢ (x ∈ {A, B} ↔ (x = A ∨ x = B))
4	2	elpw 1801	. . . . 5 ⊢ (x ∈ ℘C ↔ x ⊆ C)
5	3, 4	imbi12i 163	. . . 4 ⊢ ((x ∈ {A, B} → x ∈ ℘C) ↔ ((x = A ∨ x = B) → x ⊆ C))
6		jaob 328	. . . 4 ⊢ (((x = A ∨ x = B) → x ⊆ C) ↔ ((x = A → x ⊆ C) ∧ (x = B → x ⊆ C)))
7	5, 6	bitr 151	. . 3 ⊢ ((x ∈ {A, B} → x ∈ ℘C) ↔ ((x = A → x ⊆ C) ∧ (x = B → x ⊆ C)))
8	7	bial 695	. 2 ⊢ (∀x(x ∈ {A, B} → x ∈ ℘C) ↔ ∀x((x = A → x ⊆ C) ∧ (x = B → x ⊆ C)))
9		19.26 749	. . 3 ⊢ (∀x((x = A → x ⊆ C) ∧ (x = B → x ⊆ C)) ↔ (∀x(x = A → x ⊆ C) ∧ ∀x(x = B → x ⊆ C)))
10		prsspw.1	. . . . 5 ⊢ A ∈ V
11		sseq1 1521	. . . . 5 ⊢ (x = A → (x ⊆ C ↔ A ⊆ C))
12	10, 11	ceqsalv 1364	. . . 4 ⊢ (∀x(x = A → x ⊆ C) ↔ A ⊆ C)
13		prsspw.2	. . . . 5 ⊢ B ∈ V
14		sseq1 1521	. . . . 5 ⊢ (x = B → (x ⊆ C ↔ B ⊆ C))
15	13, 14	ceqsalv 1364	. . . 4 ⊢ (∀x(x = B → x ⊆ C) ↔ B ⊆ C)
16	12, 15	anbi12i 369	. . 3 ⊢ ((∀x(x = A → x ⊆ C) ∧ ∀x(x = B → x ⊆ C)) ↔ (A ⊆ C ∧ B ⊆ C))
17	9, 16	bitr 151	. 2 ⊢ (∀x((x = A → x ⊆ C) ∧ (x = B → x ⊆ C)) ↔ (A ⊆ C ∧ B ⊆ C))
18	1, 8, 17	3bitr 155	1 ⊢ ({A, B} ⊆ ℘C ↔ (A ⊆ C ∧ B ⊆ C))

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798  {cpr 1809

This theorem is referenced by:  dfchj3 5326

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-pw 1799  df-sn 1811  df-pr 1812

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