Theorem opidORIG 34109

Description: Please delete when opidg 41297 moves from a Mathbox to the main set.mm . (Contributed by Peter Mazsa, 22-Jul-2019.)

Assertion

Ref	Expression
opidORIG	|- ( A e. V -> <. A , A >. = { { A } } )

Proof of Theorem opidORIG

Step	Hyp	Ref	Expression
1		dfopg 4400	. . 3 |- ( ( A e. V /\ A e. V ) -> <. A , A >. = { { A } , { A , A } } )
2	1	anidms 677	. 2 |- ( A e. V -> <. A , A >. = { { A } , { A , A } } )
3		dfsn2 4190	. . . . 5 |- { A } = { A , A }
4	3	eqcomi 2631	. . . 4 |- { A , A } = { A }
5	4	preq2i 4272	. . 3 |- { { A } , { A , A } } = { { A } , { A } }
6		dfsn2 4190	. . 3 |- { { A } } = { { A } , { A } }
7	5, 6	eqtr4i 2647	. 2 |- { { A } , { A , A } } = { { A } }
8	2, 7	syl6eq 2672	1 |- ( A e. V -> <. A , A >. = { { A } } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {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:  opideq  34110