Theorem sossfld 5460

Description: The base set of a strict order is contained in the field of the relation, except possibly for one element (note that (/) Or { B }). (Contributed by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
sossfld	|- ( ( R Or A /\ B e. A ) -> ( A \ { B } ) C_ ( dom R u. ran R ) )

Proof of Theorem sossfld

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsn 4158	. . 3 |- ( x e. ( A \ { B } ) <-> ( x e. A /\ x =/= B ) )
2		sotrieq 4833	. . . . . . 7 |- ( ( R Or A /\ ( x e. A /\ B e. A ) ) -> ( x = B <-> -. ( x R B \/ B R x ) ) )
3	2	necon2abid 2721	. . . . . 6 |- ( ( R Or A /\ ( x e. A /\ B e. A ) ) -> ( ( x R B \/ B R x ) <-> x =/= B ) )
4	3	anass1rs 805	. . . . 5 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( ( x R B \/ B R x ) <-> x =/= B ) )
5		breldmg 5214	. . . . . . . . . 10 |- ( ( x e. A /\ B e. A /\ x R B ) -> x e. dom R )
6	5	3expia 1198	. . . . . . . . 9 |- ( ( x e. A /\ B e. A ) -> ( x R B -> x e. dom R ) )
7	6	adantll 713	. . . . . . . 8 |- ( ( ( R Or A /\ x e. A ) /\ B e. A ) -> ( x R B -> x e. dom R ) )
8	7	an32s 802	. . . . . . 7 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( x R B -> x e. dom R ) )
9		brelrng 5238	. . . . . . . . 9 |- ( ( B e. A /\ x e. A /\ B R x ) -> x e. ran R )
10	9	3expia 1198	. . . . . . . 8 |- ( ( B e. A /\ x e. A ) -> ( B R x -> x e. ran R ) )
11	10	adantll 713	. . . . . . 7 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( B R x -> x e. ran R ) )
12	8, 11	orim12d 836	. . . . . 6 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( ( x R B \/ B R x ) -> ( x e. dom R \/ x e. ran R ) ) )
13		elun 3650	. . . . . 6 |- ( x e. ( dom R u. ran R ) <-> ( x e. dom R \/ x e. ran R ) )
14	12, 13	syl6ibr 227	. . . . 5 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( ( x R B \/ B R x ) -> x e. ( dom R u. ran R ) ) )
15	4, 14	sylbird 235	. . . 4 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( x =/= B -> x e. ( dom R u. ran R ) ) )
16	15	expimpd 603	. . 3 |- ( ( R Or A /\ B e. A ) -> ( ( x e. A /\ x =/= B ) -> x e. ( dom R u. ran R ) ) )
17	1, 16	syl5bi 217	. 2 |- ( ( R Or A /\ B e. A ) -> ( x e. ( A \ { B } ) -> x e. ( dom R u. ran R ) ) )
18	17	ssrdv 3515	1 |- ( ( R Or A /\ B e. A ) -> ( A \ { B } ) C_ ( dom R u. ran R ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   e. wcel 1767   =/= wne 2662   \ cdif 3478   u. cun 3479   C_ wss 3481   {csn 4033   class class class wbr 4453   Or wor 4805   dom cdm 5005   ran crn 5006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-po 4806  df-so 4807  df-cnv 5013  df-dm 5015  df-rn 5016

This theorem is referenced by:  sofld  5461  soex  6738