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 4412 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
2 | df-iin 4458 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
3 | 1, 2 | eqtr4i 2635 | 1 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 {cab 2596 ∀wral 2896 ∩ cint 4410 ∩ ciin 4456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-ral 2901 df-int 4411 df-iin 4458 |
This theorem is referenced by: relint 5165 intpreima 6254 ixpint 7821 firest 15916 efger 17954 rintopn 20539 intcld 20654 iundifdifd 28762 iundifdif 28763 |
Copyright terms: Public domain | W3C validator |