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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 3113 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | iotacl 6310 | . . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) | |
3 | 1, 2 | sylbi 220 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
4 | df-riota 7093 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | df-rab 3115 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
6 | 3, 4, 5 | 3eltr4g 2907 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∃!weu 2628 {cab 2776 ∃!wreu 3108 {crab 3110 ℩cio 6281 ℩crio 7092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-uni 4801 df-iota 6283 df-riota 7093 |
This theorem is referenced by: riotacl 7110 riotasbc 7111 riotaxfrd 7127 supub 8907 suplub 8908 ordtypelem3 8968 catlid 16946 catrid 16947 grplinv 18144 pj1id 18817 evlsval2 20759 ig1pval3 24775 coelem 24823 quotlem 24896 mircgr 26451 mirbtwn 26452 grpoidinv2 28298 grpoinv 28308 cnlnadjlem5 29854 cvmsiota 32637 cvmliftiota 32661 mpaalem 40096 disjinfi 41820 |
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