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 3147 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | simpr 487 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜑) | |
3 | 2 | ss2abi 4043 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝜑} |
4 | 1, 3 | eqsstri 4001 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∈ wcel 2110 {cab 2799 {crab 3142 ⊆ wss 3936 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-in 3943 df-ss 3952 |
This theorem is referenced by: epse 5533 riotasbc 7126 toponsspwpw 21524 dmtopon 21525 aannenlem2 24912 aalioulem2 24916 ballotlemfmpn 31747 rencldnfilem 39410 rababg 39926 |
Copyright terms: Public domain | W3C validator |