![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ssextss | GIF version |
Description: An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.) |
Ref | Expression |
---|---|
ssextss | ⊢ (A ⊆ B ↔ ∀x(x ⊆ A → x ⊆ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspwb 3943 | . 2 ⊢ (A ⊆ B ↔ 𝒫 A ⊆ 𝒫 B) | |
2 | dfss2 2928 | . 2 ⊢ (𝒫 A ⊆ 𝒫 B ↔ ∀x(x ∈ 𝒫 A → x ∈ 𝒫 B)) | |
3 | vex 2554 | . . . . 5 ⊢ x ∈ V | |
4 | 3 | elpw 3357 | . . . 4 ⊢ (x ∈ 𝒫 A ↔ x ⊆ A) |
5 | 3 | elpw 3357 | . . . 4 ⊢ (x ∈ 𝒫 B ↔ x ⊆ B) |
6 | 4, 5 | imbi12i 228 | . . 3 ⊢ ((x ∈ 𝒫 A → x ∈ 𝒫 B) ↔ (x ⊆ A → x ⊆ B)) |
7 | 6 | albii 1356 | . 2 ⊢ (∀x(x ∈ 𝒫 A → x ∈ 𝒫 B) ↔ ∀x(x ⊆ A → x ⊆ B)) |
8 | 1, 2, 7 | 3bitri 195 | 1 ⊢ (A ⊆ B ↔ ∀x(x ⊆ A → x ⊆ B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 ∈ wcel 1390 ⊆ wss 2911 𝒫 cpw 3351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 |
This theorem is referenced by: ssext 3948 nssssr 3949 |
Copyright terms: Public domain | W3C validator |