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Theorem grusn 10026
 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 4455 . 2 {𝐴} = {𝐴, 𝐴}
2 grupr 10019 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐴𝑈) → {𝐴, 𝐴} ∈ 𝑈)
323anidm23 1401 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴, 𝐴} ∈ 𝑈)
41, 3syl5eqel 2870 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   ∈ wcel 2050  {csn 4442  {cpr 4444  Univcgru 10012 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-tr 5032  df-iota 6154  df-fv 6198  df-ov 6981  df-gru 10013 This theorem is referenced by:  gruop  10027  grusucd  39941
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