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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 3375 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | iotacl 6528 | . . 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 7367 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | df-rab 3431 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
6 | 3, 4, 5 | 3eltr4g 2848 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2104 ∃!weu 2560 {cab 2707 ∃!wreu 3372 {crab 3430 ℩cio 6492 ℩crio 7366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-un 3952 df-in 3954 df-ss 3964 df-sn 4628 df-pr 4630 df-uni 4908 df-iota 6494 df-riota 7367 |
This theorem is referenced by: riotacl 7385 riotasbc 7386 riotaxfrd 7402 supub 9456 suplub 9457 ordtypelem3 9517 catlid 17631 catrid 17632 grplinv 18910 pj1id 19608 evlsval2 21869 ig1pval3 25927 coelem 25975 quotlem 26049 mircgr 28175 mirbtwn 28176 grpoidinv2 30035 grpoinv 30045 cnlnadjlem5 31591 cvmsiota 34566 cvmliftiota 34590 mpaalem 42196 disjinfi 44189 |
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