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Theorem grusn 9832
Description: A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
grusn ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)

Proof of Theorem grusn
StepHypRef Expression
1 dfsn2 4330 . 2 {𝐴} = {𝐴, 𝐴}
2 grupr 9825 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐴𝑈) → {𝐴, 𝐴} ∈ 𝑈)
323anidm23 1531 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴, 𝐴} ∈ 𝑈)
41, 3syl5eqel 2854 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  {csn 4317  {cpr 4319  Univcgru 9818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-tr 4888  df-iota 5993  df-fv 6038  df-ov 6799  df-gru 9819
This theorem is referenced by:  gruop  9833
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