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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 3177 | . . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉 | |
2 | 1 | biantru 300 | . 2 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ∧ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉)) |
3 | eqss 3107 | . 2 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ∧ {𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ 𝑉)) | |
4 | 2, 3 | bitr4i 186 | 1 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 {crab 2418 ⊆ wss 3066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rab 2423 df-in 3072 df-ss 3079 |
This theorem is referenced by: difrab0eqim 3424 |
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