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Mirrors > Home > MPE Home > Th. List > ssext | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ssext | ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵)) |
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)) |
Colors of variables: wff setvar class |
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 37374 |
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