Theorem grusn 9626

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

Assertion

Ref	Expression
grusn	|- ( ( U e. Univ /\ A e. U ) -> { A } e. U )

Proof of Theorem grusn

Step	Hyp	Ref	Expression
1		dfsn2 4190	. 2 |- { A } = { A , A }
2		grupr 9619	. . 3 |- ( ( U e. Univ /\ A e. U /\ A e. U ) -> { A , A } e. U )
3	2	3anidm23 1385	. 2 |- ( ( U e. Univ /\ A e. U ) -> { A , A } e. U )
4	1, 3	syl5eqel 2705	1 |- ( ( U e. Univ /\ A e. U ) -> { A } e. U )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   {csn 4177   {cpr 4179   Univcgru 9612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-tr 4753  df-iota 5851  df-fv 5896  df-ov 6653  df-gru 9613

This theorem is referenced by:  gruop  9627