icoreclin - Mathbox for ML

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Theorem icoreclin 33205

Description: The set of closed-below, open-above intervals of reals is closed under finite intersection. (Contributed by ML, 27-Jul-2020.)

**Hypothesis**
Ref	Expression
isbasisrelowl.1

**Assertion**
Ref	Expression
icoreclin	$/\$

Distinct variable group:

Proof of Theorem icoreclin Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isbasisrelowl.1	. . . 4
2	1	icoreelrnab 33202	. . 3 $/\$
3	1	icoreelrnab 33202	. . . . . . 7 $/\$
4	1	isbasisrelowllem1 33203	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5	4	ex 450	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6	1	isbasisrelowllem2 33204	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7	6	ex 450	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8	5, 7	jaod 395	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
9		incom 3805	. . . . . . . . . . . . . . 15
10	1	isbasisrelowllem2 33204	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11	9, 10	syl5eqelr 2706	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12	11	ancom1s 847	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13	12	ex 450	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14	1	isbasisrelowllem1 33203	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	9, 14	syl5eqelr 2706	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	15	ancom1s 847	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17	16	ex 450	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	13, 17	jaod 395	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
19		3simpa 1058	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
20		3simpa 1058	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
21		letric 10137	. . . . . . . . . . . . . . 15 $/\$ $\/$
22		letric 10137	. . . . . . . . . . . . . . 15 $/\$ $\/$
23	21, 22	anim12i 590	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$
24		anddi 914	. . . . . . . . . . . . . 14 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
25	23, 24	sylib 208	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	25	an4s 869	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
27	19, 20, 26	syl2an 494	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28	8, 18, 27	mpjaod 396	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	28	ex 450	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	29	3expia 1267	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
31	30	rexlimivv 3036	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	3, 31	sylbi 207	. . . . . 6 $/\$ $/\$ $/\$
33	32	com12 32	. . . . 5 $/\$ $/\$ $/\$
34	33	3expia 1267	. . . 4 $/\$ $/\$
35	34	rexlimivv 3036	. . 3 $/\$
36	2, 35	sylbi 207	. 2
37	36	impcom 446	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wrex 2913

crab 2916

cin 3573 class class class wbr 4653

cxp 5112

cima 5117

cr 9935

clt 10074

cle 10075

cico 12177

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 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011

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-nel 2898 df-ral 2917 df-rex 2918 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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-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-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ico 12181

This theorem is referenced by: isbasisrelowl 33206

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