Theorem ssextss 5420

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 5416	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2		df-ss 3921	. 2 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
3		velpw 4560	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4		velpw 4560	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
5	3, 4	imbi12i 352	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
6	5	albii 1839	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
7	1, 2, 6	3bitri 299	1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1558   ∈ wcel 2142   ⊆ wss 3904  𝒫 cpw 4555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-ss 3921  df-pw 4557  df-sn 4583  df-pr 4585

This theorem is referenced by:  ssext  5421  nssss  5422