Theorem sossfld 5580

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 4317	. . 3 |- ( x e. ( A \ { B } ) <-> ( x e. A /\ x =/= B ) )
2		sotrieq 5062	. . . . . . 7 |- ( ( R Or A /\ ( x e. A /\ B e. A ) ) -> ( x = B <-> -. ( x R B \/ B R x ) ) )
3	2	necon2abid 2836	. . . . . 6 |- ( ( R Or A /\ ( x e. A /\ B e. A ) ) -> ( ( x R B \/ B R x ) <-> x =/= B ) )
4	3	anass1rs 849	. . . . 5 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( ( x R B \/ B R x ) <-> x =/= B ) )
5		breldmg 5330	. . . . . . . . . 10 |- ( ( x e. A /\ B e. A /\ x R B ) -> x e. dom R )
6	5	3expia 1267	. . . . . . . . 9 |- ( ( x e. A /\ B e. A ) -> ( x R B -> x e. dom R ) )
7	6	ancoms 469	. . . . . . . 8 |- ( ( B e. A /\ x e. A ) -> ( x R B -> x e. dom R ) )
8		brelrng 5355	. . . . . . . . 9 |- ( ( B e. A /\ x e. A /\ B R x ) -> x e. ran R )
9	8	3expia 1267	. . . . . . . 8 |- ( ( B e. A /\ x e. A ) -> ( B R x -> x e. ran R ) )
10	7, 9	orim12d 883	. . . . . . 7 |- ( ( B e. A /\ x e. A ) -> ( ( x R B \/ B R x ) -> ( x e. dom R \/ x e. ran R ) ) )
11		elun 3753	. . . . . . 7 |- ( x e. ( dom R u. ran R ) <-> ( x e. dom R \/ x e. ran R ) )
12	10, 11	syl6ibr 242	. . . . . 6 |- ( ( B e. A /\ x e. A ) -> ( ( x R B \/ B R x ) -> x e. ( dom R u. ran R ) ) )
13	12	adantll 750	. . . . 5 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( ( x R B \/ B R x ) -> x e. ( dom R u. ran R ) ) )
14	4, 13	sylbird 250	. . . 4 |- ( ( ( R Or A /\ B e. A ) /\ x e. A ) -> ( x =/= B -> x e. ( dom R u. ran R ) ) )
15	14	expimpd 629	. . 3 |- ( ( R Or A /\ B e. A ) -> ( ( x e. A /\ x =/= B ) -> x e. ( dom R u. ran R ) ) )
16	1, 15	syl5bi 232	. 2 |- ( ( R Or A /\ B e. A ) -> ( x e. ( A \ { B } ) -> x e. ( dom R u. ran R ) ) )
17	16	ssrdv 3609	1 |- ( ( R Or A /\ B e. A ) -> ( A \ { B } ) C_ ( dom R u. ran R ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   e. wcel 1990   =/= wne 2794   \ cdif 3571   u. cun 3572   C_ wss 3574   {csn 4177   class class class wbr 4653   Or wor 5034   dom cdm 5114   ran crn 5115

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-ne 2795  df-ral 2917  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-po 5035  df-so 5036  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  sofld  5581  soex  7109