Theorem elpwg 1802

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

Assertion

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

Proof of Theorem elpwg

Step	Hyp	Ref	Expression
1		eleq1 1149	. . 3 ⊢ (x = A → (x ∈ ℘B ↔ A ∈ ℘B))
2		sseq1 1521	. . 3 ⊢ (x = A → (x ⊆ B ↔ A ⊆ B))
3	1, 2	bibi12d 477	. 2 ⊢ (x = A → ((x ∈ ℘B ↔ x ⊆ B) ↔ (A ∈ ℘B ↔ A ⊆ B)))
4		visset 1350	. . 3 ⊢ x ∈ V
5	4	elpw 1801	. 2 ⊢ (x ∈ ℘B ↔ x ⊆ B)
6	3, 5	vtoclg 1383	1 ⊢ (A ∈ C → (A ∈ ℘B ↔ A ⊆ B))

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ℘cpw 1798

This theorem is referenced by:  elpw2g 1803  r1rankid 3537

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-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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