|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > ssab2 | Structured version Visualization version GIF version | ||
| Description: Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.) | 
| Ref | Expression | 
|---|---|
| ssab2 | ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpl 482 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴) | |
| 2 | 1 | abssi 4070 | 1 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∈ wcel 2108 {cab 2714 ⊆ 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-8 2110 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-clel 2816 df-ss 3968 | 
| This theorem is referenced by: zfausab 5332 exss 5468 dmopabss 5929 rnopabss 5966 fabexgOLD 7961 isf32lem9 10401 psubspset 39746 psubclsetN 39938 | 
| Copyright terms: Public domain | W3C validator |