Theorem ssext 4307

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 4306	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
2		ssextss 4306	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴))
3	1, 2	anbi12i 460	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ∧ ∀𝑥(𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴)))
4		eqss 3239	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
5		albiim 1533	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵) ↔ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ∧ ∀𝑥(𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴)))
6	3, 4, 5	3bitr4i 212	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1393   = wceq 1395   ⊆ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672

This theorem is referenced by: (None)