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 4880 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
2 | df-iin 4924 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
3 | 1, 2 | eqtr4i 2849 | 1 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 {cab 2801 ∀wral 3140 ∩ cint 4878 ∩ ciin 4922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-ral 3145 df-int 4879 df-iin 4924 |
This theorem is referenced by: trint 5190 relint 5694 intpreima 6840 ixpint 8491 firest 16708 efger 18846 subdrgint 19584 rintopn 21519 intcld 21650 iundifdifd 30315 iundifdif 30316 |
Copyright terms: Public domain | W3C validator |