Theorem ssext 5454

Description: An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)

Assertion

Ref	Expression
ssext	⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssext

Step	Hyp	Ref	Expression
1		ssextss 5453	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
2		ssextss 5453	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴))
3	1, 2	anbi12i 627	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ∧ ∀𝑥(𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴)))
4		eqss 3997	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
5		albiim 1892	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵) ↔ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ∧ ∀𝑥(𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴)))
6	3, 4, 5	3bitr4i 302	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541   ⊆ wss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3953  df-in 3955  df-ss 3965  df-pw 4604  df-sn 4629  df-pr 4631

This theorem is referenced by:  extssr  37682