Theorem grusn 10838

Description: A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
grusn	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈)

Proof of Theorem grusn

Step	Hyp	Ref	Expression
1		dfsn2 4636	. 2 ⊢ {𝐴} = {𝐴, 𝐴}
2		grupr 10831	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → {𝐴, 𝐴} ∈ 𝑈)
3	2	3anidm23 1418	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴, 𝐴} ∈ 𝑈)
4	1, 3	eqeltrid 2830	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2099  {csn 4623  {cpr 4625  Univcgru 10824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-tr 5263  df-iota 6498  df-fv 6554  df-ov 7419  df-gru 10825

This theorem is referenced by:  gruop  10839  grusucd  43941