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Theorem grusn 9570
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 4161 . 2 {𝐴} = {𝐴, 𝐴}
2 grupr 9563 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐴𝑈) → {𝐴, 𝐴} ∈ 𝑈)
323anidm23 1382 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴, 𝐴} ∈ 𝑈)
41, 3syl5eqel 2702 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → {𝐴} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  {csn 4148  {cpr 4150  Univcgru 9556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-tr 4713  df-iota 5810  df-fv 5855  df-ov 6607  df-gru 9557
This theorem is referenced by:  gruop  9571
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