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Mirrors > Home > ILE Home > Th. List > intiin | 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 3638 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
2 | df-iin 3681 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
3 | 1, 2 | eqtr4i 2104 | 1 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 {cab 2067 ∀wral 2348 ∩ cint 3636 ∩ ciin 3679 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-ral 2353 df-int 3637 df-iin 3681 |
This theorem is referenced by: relint 4479 |
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