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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 3350 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | iotacl 6465 | . . 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 7293 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | df-rab 3404 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
6 | 3, 4, 5 | 3eltr4g 2854 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ∃!weu 2566 {cab 2713 ∃!wreu 3347 {crab 3403 ℩cio 6429 ℩crio 7292 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-un 3903 df-in 3905 df-ss 3915 df-sn 4574 df-pr 4576 df-uni 4853 df-iota 6431 df-riota 7293 |
This theorem is referenced by: riotacl 7311 riotasbc 7312 riotaxfrd 7328 supub 9316 suplub 9317 ordtypelem3 9377 catlid 17489 catrid 17490 grplinv 18724 pj1id 19400 evlsval2 21403 ig1pval3 25445 coelem 25493 quotlem 25566 mircgr 27307 mirbtwn 27308 grpoidinv2 29165 grpoinv 29175 cnlnadjlem5 30721 cvmsiota 33538 cvmliftiota 33562 mpaalem 41240 disjinfi 43058 |
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