|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > intiin | Structured version Visualization version GIF version | ||
| Description: Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.) | 
| Ref | Expression | 
|---|---|
| intiin | ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfint2 4947 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
| 2 | df-iin 4993 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
| 3 | 1, 2 | eqtr4i 2767 | 1 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 {cab 2713 ∀wral 3060 ∩ cint 4945 ∩ 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-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-ral 3061 df-int 4946 df-iin 4993 | 
| This theorem is referenced by: trint 5276 relint 5828 intpreima 7089 ixpint 8966 firest 17478 efger 19737 subdrgint 20805 rintopn 22916 intcld 23049 iundifdifd 32575 iundifdif 32576 | 
| Copyright terms: Public domain | W3C validator |