Theorem ssext 5427

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 5426	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
2		ssextss 5426	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴))
3	1, 2	anbi12i 640	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ∧ ∀𝑥(𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴)))
4		eqss 3928	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
5		albiim 1924	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵) ↔ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ∧ ∀𝑥(𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴)))
6	3, 4, 5	3bitr4i 306	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ⊆ wss 3900

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-pw 4562  df-sn 4589  df-pr 4591

This theorem is used by:  extssr  39724