![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > caovcl | GIF version |
Description: Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.) |
Ref | Expression |
---|---|
caovcl.1 | ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
caovcl | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1336 | . 2 ⊢ ⊤ | |
2 | caovcl.1 | . . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) | |
3 | 2 | adantl 275 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
4 | 3 | caovclg 5931 | . 2 ⊢ ((⊤ ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝑆) |
5 | 1, 4 | mpan 421 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ⊤wtru 1333 ∈ wcel 1481 (class class class)co 5782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 |
This theorem is referenced by: ecopovtrn 6534 ecopovtrng 6537 genpelvl 7344 genpelvu 7345 genpml 7349 genpmu 7350 genprndl 7353 genprndu 7354 genpassl 7356 genpassu 7357 genpassg 7358 expcllem 10335 |
Copyright terms: Public domain | W3C validator |