Theorem ssextss 4318

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 4314	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2		ssalel 3216	. 2 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
3		vex 2806	. . . . 5 ⊢ 𝑥 ∈ V
4	3	elpw 3662	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
5	3	elpw 3662	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
6	4, 5	imbi12i 239	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
7	6	albii 1519	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵) ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
8	1, 2, 7	3bitri 206	1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396   ∈ wcel 2202   ⊆ wss 3201  𝒫 cpw 3656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679

This theorem is referenced by:  ssext  4319  nssssr  4320