Theorem ssextss 3947

Description: An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)

Assertion

Ref	Expression
ssextss	⊢ (A ⊆ B ↔ ∀x(x ⊆ A → x ⊆ B))

Distinct variable groups:   x,A   x,B

Proof of Theorem ssextss

Step	Hyp	Ref	Expression
1		sspwb 3943	. 2 ⊢ (A ⊆ B ↔ 𝒫 A ⊆ 𝒫 B)
2		dfss2 2928	. 2 ⊢ (𝒫 A ⊆ 𝒫 B ↔ ∀x(x ∈ 𝒫 A → x ∈ 𝒫 B))
3		vex 2554	. . . . 5 ⊢ x ∈ V
4	3	elpw 3357	. . . 4 ⊢ (x ∈ 𝒫 A ↔ x ⊆ A)
5	3	elpw 3357	. . . 4 ⊢ (x ∈ 𝒫 B ↔ x ⊆ B)
6	4, 5	imbi12i 228	. . 3 ⊢ ((x ∈ 𝒫 A → x ∈ 𝒫 B) ↔ (x ⊆ A → x ⊆ B))
7	6	albii 1356	. 2 ⊢ (∀x(x ∈ 𝒫 A → x ∈ 𝒫 B) ↔ ∀x(x ⊆ A → x ⊆ B))
8	1, 2, 7	3bitri 195	1 ⊢ (A ⊆ B ↔ ∀x(x ⊆ A → x ⊆ B))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   ∈ wcel 1390   ⊆ wss 2911  𝒫 cpw 3351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373

This theorem is referenced by:  ssext  3948  nssssr  3949