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Mirrors > Home > ILE Home > Th. List > ssrabeq | GIF version |
Description: If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) |
Ref | Expression |
---|---|
ssrabeq | ⊢ (𝑉 ⊆ {x ∈ 𝑉 ∣ φ} ↔ 𝑉 = {x ∈ 𝑉 ∣ φ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3019 | . . 3 ⊢ {x ∈ 𝑉 ∣ φ} ⊆ 𝑉 | |
2 | 1 | biantru 286 | . 2 ⊢ (𝑉 ⊆ {x ∈ 𝑉 ∣ φ} ↔ (𝑉 ⊆ {x ∈ 𝑉 ∣ φ} ∧ {x ∈ 𝑉 ∣ φ} ⊆ 𝑉)) |
3 | eqss 2954 | . 2 ⊢ (𝑉 = {x ∈ 𝑉 ∣ φ} ↔ (𝑉 ⊆ {x ∈ 𝑉 ∣ φ} ∧ {x ∈ 𝑉 ∣ φ} ⊆ 𝑉)) | |
4 | 2, 3 | bitr4i 176 | 1 ⊢ (𝑉 ⊆ {x ∈ 𝑉 ∣ φ} ↔ 𝑉 = {x ∈ 𝑉 ∣ φ}) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1242 {crab 2304 ⊆ wss 2911 |
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-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rab 2309 df-in 2918 df-ss 2925 |
This theorem is referenced by: difrab0eqim 3282 |
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