Theorem grusn 10228

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 4582	. 2 ⊢ {𝐴} = {𝐴, 𝐴}
2		grupr 10221	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → {𝐴, 𝐴} ∈ 𝑈)
3	2	3anidm23 1417	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴, 𝐴} ∈ 𝑈)
4	1, 3	eqeltrid 2919	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  {csn 4569  {cpr 4571  Univcgru 10214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-tr 5175  df-iota 6316  df-fv 6365  df-ov 7161  df-gru 10215

This theorem is referenced by:  gruop  10229  grusucd  40573