ordtbaslem - Metamath Proof Explorer

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Theorem ordtbaslem 20992

Description: Lemma for ordtbas 20996. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Hypotheses**
Ref	Expression
ordtval.1
ordtval.2

**Assertion**
Ref	Expression
ordtbaslem

Distinct variable groups:

Allowed substitution hints:

(

)

Proof of Theorem ordtbaslem Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3anrot 1043	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
2		ordtval.1	. . . . . . . . . . . . . 14
3	2	tsrlemax 17220	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $\/$
4	1, 3	sylan2br 493	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $\/$
5	4	3exp2 1285	. . . . . . . . . . 11 $\/$
6	5	imp42 620	. . . . . . . . . 10 $/\$ $/\$ $/\$ $\/$
7	6	notbid 308	. . . . . . . . 9 $/\$ $/\$ $/\$ $\/$
8		ioran 511	. . . . . . . . 9 $\/$ $/\$
9	7, 8	syl6bb 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
10	9	rabbidva 3188	. . . . . . 7 $/\$ $/\$ $/\$
11		ifcl 4130	. . . . . . . . . 10 $/\$
12	11	ancoms 469	. . . . . . . . 9 $/\$
13	12	adantl 482	. . . . . . . 8 $/\$ $/\$
14		dmexg 7097	. . . . . . . . . . . 12
15	2, 14	syl5eqel 2705	. . . . . . . . . . 11
16	15	adantr 481	. . . . . . . . . 10 $/\$ $/\$
17		rabexg 4812	. . . . . . . . . 10 $/\$
18	16, 17	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$
19	10, 18	eqeltrd 2701	. . . . . . . 8 $/\$ $/\$
20		eqid 2622	. . . . . . . . . 10
21		breq2 4657	. . . . . . . . . . . 12
22	21	notbid 308	. . . . . . . . . . 11
23	22	rabbidv 3189	. . . . . . . . . 10
24	20, 23	elrnmpt1s 5373	. . . . . . . . 9 $/\$
25		ordtval.2	. . . . . . . . 9
26	24, 25	syl6eleqr 2712	. . . . . . . 8 $/\$
27	13, 19, 26	syl2anc 693	. . . . . . 7 $/\$ $/\$
28	10, 27	eqeltrrd 2702	. . . . . 6 $/\$ $/\$ $/\$
29	28	ralrimivva 2971	. . . . 5 $/\$
30		rabexg 4812	. . . . . . . 8
31	15, 30	syl 17	. . . . . . 7
32	31	ralrimivw 2967	. . . . . 6
33		breq2 4657	. . . . . . . . . 10
34	33	notbid 308	. . . . . . . . 9
35	34	rabbidv 3189	. . . . . . . 8
36	35	cbvmptv 4750	. . . . . . 7
37		ineq1 3807	. . . . . . . . . 10
38		inrab 3899	. . . . . . . . . 10 $/\$
39	37, 38	syl6eq 2672	. . . . . . . . 9 $/\$
40	39	eleq1d 2686	. . . . . . . 8 $/\$
41	40	ralbidv 2986	. . . . . . 7 $/\$
42	36, 41	ralrnmpt 6368	. . . . . 6 $/\$
43	32, 42	syl 17	. . . . 5 $/\$
44	29, 43	mpbird 247	. . . 4
45		rabexg 4812	. . . . . . . 8
46	15, 45	syl 17	. . . . . . 7
47	46	ralrimivw 2967	. . . . . 6
48		breq2 4657	. . . . . . . . . 10
49	48	notbid 308	. . . . . . . . 9
50	49	rabbidv 3189	. . . . . . . 8
51	50	cbvmptv 4750	. . . . . . 7
52		ineq2 3808	. . . . . . . 8
53	52	eleq1d 2686	. . . . . . 7
54	51, 53	ralrnmpt 6368	. . . . . 6
55	47, 54	syl 17	. . . . 5
56	55	ralbidv 2986	. . . 4
57	44, 56	mpbird 247	. . 3
58	25	raleqi 3142	. . . 4
59	25, 58	raleqbii 2990	. . 3
60	57, 59	sylibr 224	. 2
61		pwexg 4850	. . . . 5
62	15, 61	syl 17	. . . 4
63		ssrab2 3687	. . . . . . . 8
64	15	adantr 481	. . . . . . . . 9 $/\$
65		elpw2g 4827	. . . . . . . . 9
66	64, 65	syl 17	. . . . . . . 8 $/\$
67	63, 66	mpbiri 248	. . . . . . 7 $/\$
68	67, 20	fmptd 6385	. . . . . 6
69		frn 6053	. . . . . 6
70	68, 69	syl 17	. . . . 5
71	25, 70	syl5eqss 3649	. . . 4
72	62, 71	ssexd 4805	. . 3
73		inficl 8331	. . 3
74	72, 73	syl 17	. 2
75	60, 74	mpbid 222	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

crab 2916

cvv 3200

cin 3573

wss 3574

cif 4086

cpw 4158 class class class wbr 4653

cmpt 4729

cdm 5114

crn 5115

wf 5884

cfv 5888

cfi 8316

ctsr 17199

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-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-ps 17200 df-tsr 17201

This theorem is referenced by: ordtbas2 20995

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