![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > difrab0eq | Structured version Visualization version GIF version |
Description: If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) |
Ref | Expression |
---|---|
difrab0eq | ⊢ ((𝑉 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅ ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdif0 4372 | . 2 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅) | |
2 | ssrabeq 4094 | . 2 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑}) | |
3 | 1, 2 | bitr3i 277 | 1 ⊢ ((𝑉 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅ ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 {crab 3433 ∖ cdif 3960 ⊆ wss 3963 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-ss 3980 df-nul 4340 |
This theorem is referenced by: frgrregorufr0 30353 |
Copyright terms: Public domain | W3C validator |