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Mirrors > Home > ILE Home > Th. List > riotacl | GIF version |
Description: Closure of restricted iota. (Contributed by NM, 21-Aug-2011.) |
Ref | Expression |
---|---|
riotacl | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3182 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
2 | riotacl2 5743 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
3 | 1, 2 | sseldi 3095 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∃!wreu 2418 {crab 2420 ℩crio 5729 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-uni 3737 df-iota 5088 df-riota 5730 |
This theorem is referenced by: riotaprop 5753 riotass2 5756 riotass 5757 acexmidlemcase 5769 supclti 6885 caucvgsrlemcl 7604 caucvgsrlemgt1 7610 axcaucvglemcl 7710 subval 7961 subcl 7968 divvalap 8441 divclap 8445 lbcl 8711 divfnzn 9420 flqcl 10053 flapcl 10055 cjval 10624 cjth 10625 cjf 10626 oddpwdclemodd 11857 oddpwdclemdc 11858 oddpwdc 11859 qnumdencl 11872 qnumdenbi 11877 |
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