Theorem elpw2g 1803

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47.

Assertion

Ref	Expression
elpw2g	⊢ (B ∈ C → (A ∈ ℘B ↔ A ⊆ B))

Proof of Theorem elpw2g

Step	Hyp	Ref	Expression
1		elpwg 1802	. . . 4 ⊢ (A ∈ ℘B → (A ∈ ℘B ↔ A ⊆ B))
2	1	ibi 449	. . 3 ⊢ (A ∈ ℘B → A ⊆ B)
3	2	a1i 7	. 2 ⊢ (B ∈ C → (A ∈ ℘B → A ⊆ B))
4		id 9	. . 3 ⊢ (A ⊆ B → A ⊆ B)
5		ssexg 1702	. . . . 5 ⊢ (B ∈ C → (A ⊆ B → A ∈ V))
6		elpwg 1802	. . . . 5 ⊢ (A ∈ V → (A ∈ ℘B ↔ A ⊆ B))
7	5, 6	syl6 23	. . . 4 ⊢ (B ∈ C → (A ⊆ B → (A ∈ ℘B ↔ A ⊆ B)))
8	7	pm5.74d 444	. . 3 ⊢ (B ∈ C → ((A ⊆ B → A ∈ ℘B) ↔ (A ⊆ B → A ⊆ B)))
9	4, 8	mpbiri 169	. 2 ⊢ (B ∈ C → (A ⊆ B → A ∈ ℘B))
10	3, 9	impbid 397	1 ⊢ (B ∈ C → (A ∈ ℘B ↔ A ⊆ B))

Syntax hints:   → wi 2   ↔ wb 127   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798

This theorem is referenced by:  rankval2 3514  rankss 3531  aceq3lem 3555  ocvalt 5161  spanvalt 5300  hsupval2t 5301  sshjvalt 5321  sshjval3t 5327

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-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-pw 1799

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