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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 3240 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
2 | riotacl2 5838 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
3 | 1, 2 | sselid 3153 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ∃!wreu 2457 {crab 2459 ℩crio 5824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-sn 3597 df-pr 3598 df-uni 3808 df-iota 5174 df-riota 5825 |
This theorem is referenced by: riotaprop 5848 riotass2 5851 riotass 5852 acexmidlemcase 5864 supclti 6991 caucvgsrlemcl 7776 caucvgsrlemgt1 7782 axcaucvglemcl 7882 subval 8136 subcl 8143 divvalap 8617 divclap 8621 lbcl 8889 divfnzn 9607 flqcl 10256 flapcl 10258 cjval 10835 cjth 10836 cjf 10837 oddpwdclemodd 12152 oddpwdclemdc 12153 oddpwdc 12154 qnumdencl 12167 qnumdenbi 12172 ismgmid 12685 grpinvf 12807 |
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