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Mirrors > Home > MPE Home > Th. List > rabssab | Structured version Visualization version GIF version |
Description: A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
rabssab | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3431 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | simpr 483 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜑) | |
3 | 2 | ss2abi 4063 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝜑} |
4 | 1, 3 | eqsstri 4016 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∈ wcel 2098 {cab 2705 {crab 3430 ⊆ wss 3949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3431 df-v 3475 df-in 3956 df-ss 3966 |
This theorem is referenced by: epse 5665 riotasbc 7401 cshwsexa 14814 toponsspwpw 22844 dmtopon 22845 aannenlem2 26284 aalioulem2 26288 ballotlemfmpn 34147 rencldnfilem 42271 rmxyelqirr 42361 rababg 43035 |
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