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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 3378 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | iotacl 6530 | . . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) | |
3 | 1, 2 | sylbi 216 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
4 | df-riota 7365 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | df-rab 3434 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
6 | 3, 4, 5 | 3eltr4g 2851 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ∃!weu 2563 {cab 2710 ∃!wreu 3375 {crab 3433 ℩cio 6494 ℩crio 7364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-un 3954 df-in 3956 df-ss 3966 df-sn 4630 df-pr 4632 df-uni 4910 df-iota 6496 df-riota 7365 |
This theorem is referenced by: riotacl 7383 riotasbc 7384 riotaxfrd 7400 supub 9454 suplub 9455 ordtypelem3 9515 catlid 17627 catrid 17628 grplinv 18874 pj1id 19567 evlsval2 21650 ig1pval3 25692 coelem 25740 quotlem 25813 mircgr 27908 mirbtwn 27909 grpoidinv2 29768 grpoinv 29778 cnlnadjlem5 31324 cvmsiota 34268 cvmliftiota 34292 mpaalem 41894 disjinfi 43891 |
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