Theorem sossfld 5305

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 4110	. . 3 |- ( x e. ( A \ { B } ) <-> ( x e. A /\ x =/= B ) )
2		sotrieq 4804	. . . . . . 7 |- ( ( R Or A /\ ( x e. A /\ B e. A ) ) -> ( x = B <-> -. ( x R B \/ B R x ) ) )
3	2	necon2abid 2678	. . . . . 6 |- ( ( R Or A /\ ( x e. A /\ B e. A ) ) -> ( ( x R B \/ B R x ) <-> x =/= B ) )
4	3	anass1rs 821	. . . . 5 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( ( x R B \/ B R x ) <-> x =/= B ) )
5		breldmg 5062	. . . . . . . . . 10 |- ( ( x e. A /\ B e. A /\ x R B ) -> x e. dom R )
6	5	3expia 1217	. . . . . . . . 9 |- ( ( x e. A /\ B e. A ) -> ( x R B -> x e. dom R ) )
7	6	adantll 725	. . . . . . . 8 |- ( ( ( R Or A /\ x e. A ) /\ B e. A ) -> ( x R B -> x e. dom R ) )
8	7	an32s 818	. . . . . . 7 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( x R B -> x e. dom R ) )
9		brelrng 5086	. . . . . . . . 9 |- ( ( B e. A /\ x e. A /\ B R x ) -> x e. ran R )
10	9	3expia 1217	. . . . . . . 8 |- ( ( B e. A /\ x e. A ) -> ( B R x -> x e. ran R ) )
11	10	adantll 725	. . . . . . 7 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( B R x -> x e. ran R ) )
12	8, 11	orim12d 854	. . . . . 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 3586	. . . . . 6 |- ( x e. ( dom R u. ran R ) <-> ( x e. dom R \/ x e. ran R ) )
14	12, 13	syl6ibr 235	. . . . 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 243	. . . 4 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( x =/= B -> x e. ( dom R u. ran R ) ) )
16	15	expimpd 612	. . 3 |- ( ( R Or A /\ B e. A ) -> ( ( x e. A /\ x =/= B ) -> x e. ( dom R u. ran R ) ) )
17	1, 16	syl5bi 225	. 2 |- ( ( R Or A /\ B e. A ) -> ( x e. ( A \ { B } ) -> x e. ( dom R u. ran R ) ) )
18	17	ssrdv 3450	1 |- ( ( R Or A /\ B e. A ) -> ( A \ { B } ) C_ ( dom R u. ran R ) )

Syntax hints:   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   e. wcel 1898   =/= wne 2633   \ cdif 3413   u. cun 3414   C_ wss 3416   {csn 3980   class class class wbr 4418   Or wor 4776   dom cdm 4856   ran crn 4857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419  df-opab 4478  df-po 4777  df-so 4778  df-cnv 4864  df-dm 4866  df-rn 4867

This theorem is referenced by:  sofld  5306  soex  6768