Theorem ordtt1 21183

Description: The order topology is T₁ for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
ordtt1	|- ( R e. PosetRel -> (ordTop ` R ) e. Fre )

Proof of Theorem ordtt1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordttop 21004	. 2 |- ( R e. PosetRel -> (ordTop ` R ) e. Top )
2		snssi 4339	. . . . . . . 8 |- ( x e. dom R -> { x } C_ dom R )
3	2	adantl 482	. . . . . . 7 |- ( ( R e. PosetRel /\ x e. dom R ) -> { x } C_ dom R )
4		sseqin2 3817	. . . . . . 7 |- ( { x } C_ dom R <-> ( dom R i^i { x } ) = { x } )
5	3, 4	sylib 208	. . . . . 6 |- ( ( R e. PosetRel /\ x e. dom R ) -> ( dom R i^i { x } ) = { x } )
6		velsn 4193	. . . . . . . 8 |- ( y e. { x } <-> y = x )
7		eqid 2622	. . . . . . . . . . . . 13 |- dom R = dom R
8	7	psref 17208	. . . . . . . . . . . 12 |- ( ( R e. PosetRel /\ x e. dom R ) -> x R x )
9	8	adantr 481	. . . . . . . . . . 11 |- ( ( ( R e. PosetRel /\ x e. dom R ) /\ y e. dom R ) -> x R x )
10	9, 9	jca 554	. . . . . . . . . 10 |- ( ( ( R e. PosetRel /\ x e. dom R ) /\ y e. dom R ) -> ( x R x /\ x R x ) )
11		breq2 4657	. . . . . . . . . . 11 |- ( y = x -> ( x R y <-> x R x ) )
12		breq1 4656	. . . . . . . . . . 11 |- ( y = x -> ( y R x <-> x R x ) )
13	11, 12	anbi12d 747	. . . . . . . . . 10 |- ( y = x -> ( ( x R y /\ y R x ) <-> ( x R x /\ x R x ) ) )
14	10, 13	syl5ibrcom 237	. . . . . . . . 9 |- ( ( ( R e. PosetRel /\ x e. dom R ) /\ y e. dom R ) -> ( y = x -> ( x R y /\ y R x ) ) )
15		psasym 17210	. . . . . . . . . . . 12 |- ( ( R e. PosetRel /\ x R y /\ y R x ) -> x = y )
16	15	eqcomd 2628	. . . . . . . . . . 11 |- ( ( R e. PosetRel /\ x R y /\ y R x ) -> y = x )
17	16	3expib 1268	. . . . . . . . . 10 |- ( R e. PosetRel -> ( ( x R y /\ y R x ) -> y = x ) )
18	17	ad2antrr 762	. . . . . . . . 9 |- ( ( ( R e. PosetRel /\ x e. dom R ) /\ y e. dom R ) -> ( ( x R y /\ y R x ) -> y = x ) )
19	14, 18	impbid 202	. . . . . . . 8 |- ( ( ( R e. PosetRel /\ x e. dom R ) /\ y e. dom R ) -> ( y = x <-> ( x R y /\ y R x ) ) )
20	6, 19	syl5bb 272	. . . . . . 7 |- ( ( ( R e. PosetRel /\ x e. dom R ) /\ y e. dom R ) -> ( y e. { x } <-> ( x R y /\ y R x ) ) )
21	20	rabbi2dva 3821	. . . . . 6 |- ( ( R e. PosetRel /\ x e. dom R ) -> ( dom R i^i { x } ) = { y e. dom R | ( x R y /\ y R x ) } )
22	5, 21	eqtr3d 2658	. . . . 5 |- ( ( R e. PosetRel /\ x e. dom R ) -> { x } = { y e. dom R | ( x R y /\ y R x ) } )
23	7	ordtcld3 21003	. . . . . 6 |- ( ( R e. PosetRel /\ x e. dom R /\ x e. dom R ) -> { y e. dom R | ( x R y /\ y R x ) } e. ( Clsd ` (ordTop ` R ) ) )
24	23	3anidm23 1385	. . . . 5 |- ( ( R e. PosetRel /\ x e. dom R ) -> { y e. dom R | ( x R y /\ y R x ) } e. ( Clsd ` (ordTop ` R ) ) )
25	22, 24	eqeltrd 2701	. . . 4 |- ( ( R e. PosetRel /\ x e. dom R ) -> { x } e. ( Clsd ` (ordTop ` R ) ) )
26	25	ralrimiva 2966	. . 3 |- ( R e. PosetRel -> A. x e. dom R { x } e. ( Clsd ` (ordTop ` R ) ) )
27	7	ordttopon 20997	. . . . 5 |- ( R e. PosetRel -> (ordTop ` R ) e. (TopOn ` dom R ) )
28		toponuni 20719	. . . . 5 |- ( (ordTop ` R ) e. (TopOn ` dom R ) -> dom R = U. (ordTop ` R ) )
29	27, 28	syl 17	. . . 4 |- ( R e. PosetRel -> dom R = U. (ordTop ` R ) )
30	29	raleqdv 3144	. . 3 |- ( R e. PosetRel -> ( A. x e. dom R { x } e. ( Clsd ` (ordTop ` R ) ) <-> A. x e. U. (ordTop ` R ) { x } e. ( Clsd ` (ordTop ` R ) ) ) )
31	26, 30	mpbid 222	. 2 |- ( R e. PosetRel -> A. x e. U. (ordTop ` R ) { x } e. ( Clsd ` (ordTop ` R ) ) )
32		eqid 2622	. . 3 |- U. (ordTop ` R ) = U. (ordTop ` R )
33	32	ist1 21125	. 2 |- ( (ordTop ` R ) e. Fre <-> ( (ordTop ` R ) e. Top /\ A. x e. U. (ordTop ` R ) { x } e. ( Clsd ` (ordTop ` R ) ) ) )
34	1, 31, 33	sylanbrc 698	1 |- ( R e. PosetRel -> (ordTop ` R ) e. Fre )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   i^i cin 3573   C_ wss 3574   {csn 4177   U.cuni 4436   class class class wbr 4653   dom cdm 5114   ` cfv 5888  ordTopcordt 16159   PosetRelcps 17198   Topctop 20698  TopOnctopon 20715   Clsdccld 20820   Frect1 21111

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-topgen 16104  df-ordt 16161  df-ps 17200  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-t1 21118

This theorem is referenced by: (None)