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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 4190 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | grupr 9619 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → {𝐴, 𝐴} ∈ 𝑈) | |
3 | 2 | 3anidm23 1385 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴, 𝐴} ∈ 𝑈) |
4 | 1, 3 | syl5eqel 2705 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 {csn 4177 {cpr 4179 Univcgru 9612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-tr 4753 df-iota 5851 df-fv 5896 df-ov 6653 df-gru 9613 |
This theorem is referenced by: gruop 9627 |
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