Theorem gruop 9627

Description: A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)

Assertion

Ref	Expression
gruop	|- ( ( U e. Univ /\ A e. U /\ B e. U ) -> <. A , B >. e. U )

Proof of Theorem gruop

Step	Hyp	Ref	Expression
1		dfopg 4400	. . 3 |- ( ( A e. U /\ B e. U ) -> <. A , B >. = { { A } , { A , B } } )
2	1	3adant1 1079	. 2 |- ( ( U e. Univ /\ A e. U /\ B e. U ) -> <. A , B >. = { { A } , { A , B } } )
3		simp1 1061	. . 3 |- ( ( U e. Univ /\ A e. U /\ B e. U ) -> U e. Univ )
4		grusn 9626	. . . 4 |- ( ( U e. Univ /\ A e. U ) -> { A } e. U )
5	4	3adant3 1081	. . 3 |- ( ( U e. Univ /\ A e. U /\ B e. U ) -> { A } e. U )
6		grupr 9619	. . 3 |- ( ( U e. Univ /\ A e. U /\ B e. U ) -> { A , B } e. U )
7		grupr 9619	. . 3 |- ( ( U e. Univ /\ { A } e. U /\ { A , B } e. U ) -> { { A } , { A , B } } e. U )
8	3, 5, 6, 7	syl3anc 1326	. 2 |- ( ( U e. Univ /\ A e. U /\ B e. U ) -> { { A } , { A , B } } e. U )
9	2, 8	eqeltrd 2701	1 |- ( ( U e. Univ /\ A e. U /\ B e. U ) -> <. A , B >. e. U )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   {csn 4177   {cpr 4179   <.cop 4183   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:  gruf  9633