Theorem somin1 5529

Description: Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
somin1	|- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> if ( A R B , A , B ) ( R u. _I ) A )

Proof of Theorem somin1

Step	Hyp	Ref	Expression
1		iftrue 4092	. . . . 5 |- ( A R B -> if ( A R B , A , B ) = A )
2	1	olcd 408	. . . 4 |- ( A R B -> ( if ( A R B , A , B ) R A \/ if ( A R B , A , B ) = A ) )
3	2	adantl 482	. . 3 |- ( ( ( R Or X /\ ( A e. X /\ B e. X ) ) /\ A R B ) -> ( if ( A R B , A , B ) R A \/ if ( A R B , A , B ) = A ) )
4		sotric 5061	. . . . . . 7 |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> ( A R B <-> -. ( A = B \/ B R A ) ) )
5		orcom 402	. . . . . . . . 9 |- ( ( A = B \/ B R A ) <-> ( B R A \/ A = B ) )
6		eqcom 2629	. . . . . . . . . 10 |- ( A = B <-> B = A )
7	6	orbi2i 541	. . . . . . . . 9 |- ( ( B R A \/ A = B ) <-> ( B R A \/ B = A ) )
8	5, 7	bitri 264	. . . . . . . 8 |- ( ( A = B \/ B R A ) <-> ( B R A \/ B = A ) )
9	8	notbii 310	. . . . . . 7 |- ( -. ( A = B \/ B R A ) <-> -. ( B R A \/ B = A ) )
10	4, 9	syl6bb 276	. . . . . 6 |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> ( A R B <-> -. ( B R A \/ B = A ) ) )
11	10	con2bid 344	. . . . 5 |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> ( ( B R A \/ B = A ) <-> -. A R B ) )
12	11	biimpar 502	. . . 4 |- ( ( ( R Or X /\ ( A e. X /\ B e. X ) ) /\ -. A R B ) -> ( B R A \/ B = A ) )
13		iffalse 4095	. . . . . 6 |- ( -. A R B -> if ( A R B , A , B ) = B )
14		breq1 4656	. . . . . . 7 |- ( if ( A R B , A , B ) = B -> ( if ( A R B , A , B ) R A <-> B R A ) )
15		eqeq1 2626	. . . . . . 7 |- ( if ( A R B , A , B ) = B -> ( if ( A R B , A , B ) = A <-> B = A ) )
16	14, 15	orbi12d 746	. . . . . 6 |- ( if ( A R B , A , B ) = B -> ( ( if ( A R B , A , B ) R A \/ if ( A R B , A , B ) = A ) <-> ( B R A \/ B = A ) ) )
17	13, 16	syl 17	. . . . 5 |- ( -. A R B -> ( ( if ( A R B , A , B ) R A \/ if ( A R B , A , B ) = A ) <-> ( B R A \/ B = A ) ) )
18	17	adantl 482	. . . 4 |- ( ( ( R Or X /\ ( A e. X /\ B e. X ) ) /\ -. A R B ) -> ( ( if ( A R B , A , B ) R A \/ if ( A R B , A , B ) = A ) <-> ( B R A \/ B = A ) ) )
19	12, 18	mpbird 247	. . 3 |- ( ( ( R Or X /\ ( A e. X /\ B e. X ) ) /\ -. A R B ) -> ( if ( A R B , A , B ) R A \/ if ( A R B , A , B ) = A ) )
20	3, 19	pm2.61dan 832	. 2 |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> ( if ( A R B , A , B ) R A \/ if ( A R B , A , B ) = A ) )
21		poleloe 5527	. . 3 |- ( A e. X -> ( if ( A R B , A , B ) ( R u. _I ) A <-> ( if ( A R B , A , B ) R A \/ if ( A R B , A , B ) = A ) ) )
22	21	ad2antrl 764	. 2 |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> ( if ( A R B , A , B ) ( R u. _I ) A <-> ( if ( A R B , A , B ) R A \/ if ( A R B , A , B ) = A ) ) )
23	20, 22	mpbird 247	1 |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> if ( A R B , A , B ) ( R u. _I ) A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   ifcif 4086   class class class wbr 4653   _I cid 5023   Or wor 5034

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  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-br 4654  df-opab 4713  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121

This theorem is referenced by:  somin2  5531  soltmin  5532