Theorem opid 4421

Description: The ordered pair <. A , A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.) (Avoid depending on this detail.)

Hypothesis

Ref	Expression
opid.1	|- A e. _V

Assertion

Ref	Expression
opid	|- <. A , A >. = { { A } }

Proof of Theorem opid

Step	Hyp	Ref	Expression
1		dfsn2 4190	. . 3 |- { A } = { A , A }
2	1	preq2i 4272	. 2 |- { { A } , { A } } = { { A } , { A , A } }
3		dfsn2 4190	. 2 |- { { A } } = { { A } , { A } }
4		opid.1	. . 3 |- A e. _V
5	4, 4	dfop 4401	. 2 |- <. A , A >. = { { A } , { A , A } }
6	2, 3, 5	3eqtr4ri 2655	1 |- <. A , A >. = { { A } }

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   {cpr 4179   <.cop 4183

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-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

This theorem is referenced by:  dmsnsnsn  5613  funopg  5922  vtxval3sn  25935  iedgval3sn  25936