|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > ssrd | Structured version Visualization version GIF version | ||
| Description: Deduction based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.) | 
| Ref | Expression | 
|---|---|
| ssrd.0 | ⊢ Ⅎ𝑥𝜑 | 
| ssrd.1 | ⊢ Ⅎ𝑥𝐴 | 
| ssrd.2 | ⊢ Ⅎ𝑥𝐵 | 
| ssrd.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | 
| Ref | Expression | 
|---|---|
| ssrd | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssrd.0 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | ssrd.3 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
| 3 | 1, 2 | alrimi 2212 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | 
| 4 | ssrd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 5 | ssrd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
| 6 | 4, 5 | dfssf 3973 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | 
| 7 | 3, 6 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 ⊆ wss 3950 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-11 2156 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-clel 2815 df-nfc 2891 df-ss 3967 | 
| This theorem is referenced by: rabss3d 4080 neiptopnei 23141 topdifinffinlem 37349 relowlssretop 37365 ralssiun 37409 sticksstones1 42148 sticksstones11 42158 ssdf2 45151 ssfiunibd 45326 stoweidlem52 46072 stoweidlem59 46079 | 
| Copyright terms: Public domain | W3C validator |