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Mirrors > Home > MPE Home > Th. List > riotacl2 | Structured version Visualization version GIF version |
Description: Membership law for "the unique element in 𝐴 such that 𝜑". (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
riotacl2 | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 3145 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | iotacl 6341 | . . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) | |
3 | 1, 2 | sylbi 219 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
4 | df-riota 7114 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | df-rab 3147 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
6 | 3, 4, 5 | 3eltr4g 2930 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∃!weu 2653 {cab 2799 ∃!wreu 3140 {crab 3142 ℩cio 6312 ℩crio 7113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-un 3941 df-in 3943 df-ss 3952 df-sn 4568 df-pr 4570 df-uni 4839 df-iota 6314 df-riota 7114 |
This theorem is referenced by: riotacl 7131 riotasbc 7132 riotaxfrd 7148 supub 8923 suplub 8924 ordtypelem3 8984 catlid 16954 catrid 16955 grplinv 18152 pj1id 18825 evlsval2 20300 ig1pval3 24768 coelem 24816 quotlem 24889 mircgr 26443 mirbtwn 26444 grpoidinv2 28292 grpoinv 28302 cnlnadjlem5 29848 cvmsiota 32524 cvmliftiota 32548 mpaalem 39772 disjinfi 41474 |
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