Theorem grusn 10873

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 4661	. 2 ⊢ {𝐴} = {𝐴, 𝐴}
2		grupr 10866	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → {𝐴, 𝐴} ∈ 𝑈)
3	2	3anidm23 1421	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴, 𝐴} ∈ 𝑈)
4	1, 3	eqeltrid 2848	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  {csn 4648  {cpr 4650  Univcgru 10859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-tr 5284  df-iota 6525  df-fv 6581  df-ov 7451  df-gru 10860

This theorem is referenced by:  gruop  10874  grusucd  44199