Theorem opid 3826

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

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 3636	. . . 4 |- { A } = { A , A }
2	1	eqcomi 2200	. . 3 |- { A , A } = { A }
3	2	preq2i 3703	. 2 |- { { A } , { A , A } } = { { A } , { A } }
4		opid.1	. . 3 |- A e. _V
5	4, 4	dfop 3807	. 2 |- <. A , A >. = { { A } , { A , A } }
6		dfsn2 3636	. 2 |- { { A } } = { { A } , { A } }
7	3, 5, 6	3eqtr4i 2227	1 |- <. A , A >. = { { A } }

Syntax hints:   = wceq 1364   e. wcel 2167   _Vcvv 2763   {csn 3622   {cpr 3623   <.cop 3625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631

This theorem is referenced by:  dmsnsnsng  5147  funopg  5292