|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 3437 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
| 2 | simpr 484 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜑) | |
| 3 | 2 | ss2abi 4067 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝜑} | 
| 4 | 1, 3 | eqsstri 4030 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∈ wcel 2108 {cab 2714 {crab 3436 ⊆ wss 3951 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-rab 3437 df-ss 3968 | 
| This theorem is referenced by: epse 5667 riotasbc 7406 cshwsexa 14862 toponsspwpw 22928 dmtopon 22929 aannenlem2 26371 aalioulem2 26375 ballotlemfmpn 34497 rencldnfilem 42831 rmxyelqirr 42921 rababg 43587 | 
| Copyright terms: Public domain | W3C validator |