|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > ssiin | Structured version Visualization version GIF version | ||
| Description: Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.) | 
| Ref | Expression | 
|---|---|
| ssiin | ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfcv 2904 | . 2 ⊢ Ⅎ𝑥𝐶 | |
| 2 | 1 | ssiinf 5053 | 1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wral 3060 ⊆ wss 3950 ∩ ciin 4991 | 
| 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-9 2117 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-v 3481 df-ss 3967 df-iin 4993 | 
| This theorem is referenced by: triin 5275 cflim2 10304 ptbasfi 23590 limciun 25930 clsint2 36331 fnemeet2 36369 dihglblem4 41300 dihglblem6 41343 iooiinicc 45560 iooiinioc 45574 iinhoiicc 46694 smfsuplem1 46831 | 
| Copyright terms: Public domain | W3C validator |