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Theorem grusn 10796
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 4634 . 2 {𝐴} = {𝐴, 𝐴}
2 grupr 10789 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐴𝑈) → {𝐴, 𝐴} ∈ 𝑈)
323anidm23 1418 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴, 𝐴} ∈ 𝑈)
41, 3eqeltrid 2829 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  {csn 4621  {cpr 4623  Univcgru 10782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-tr 5257  df-iota 6486  df-fv 6542  df-ov 7405  df-gru 10783
This theorem is referenced by:  gruop  10797  grusucd  43503
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