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Mirrors > Home > MPE Home > Th. List > grusn | Structured version Visualization version GIF version |
Description: A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
grusn | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 4636 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | grupr 10831 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → {𝐴, 𝐴} ∈ 𝑈) | |
3 | 2 | 3anidm23 1418 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴, 𝐴} ∈ 𝑈) |
4 | 1, 3 | eqeltrid 2830 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 {csn 4623 {cpr 4625 Univcgru 10824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-tr 5263 df-iota 6498 df-fv 6554 df-ov 7419 df-gru 10825 |
This theorem is referenced by: gruop 10839 grusucd 43941 |
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