Theorem sossfld 5280

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 3995	. . 3 |- ( x e. ( A \ { B } ) <-> ( x e. A /\ x =/= B ) )
2		sotrieq 4663	. . . . . . 7 |- ( ( R Or A /\ ( x e. A /\ B e. A ) ) -> ( x = B <-> -. ( x R B \/ B R x ) ) )
3	2	necon2abid 2663	. . . . . 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 5040	. . . . . . . . . 10 |- ( ( x e. A /\ B e. A /\ x R B ) -> x e. dom R )
6	5	3expia 1189	. . . . . . . . 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 5064	. . . . . . . . 9 |- ( ( B e. A /\ x e. A /\ B R x ) -> x e. ran R )
10	9	3expia 1189	. . . . . . . 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 834	. . . . . 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 3492	. . . . . 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 3357	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 1756   =/= wne 2601   \ cdif 3320   u. cun 3321   C_ wss 3323   {csn 3872   class class class wbr 4287   Or wor 4635   dom cdm 4835   ran crn 4836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-po 4636  df-so 4637  df-cnv 4843  df-dm 4845  df-rn 4846

This theorem is referenced by:  sofld  5281  soex  6516