nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4360

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4278 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6100 or nntri3or 6095), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.)

**Hypothesis**
Ref	Expression
nnregexmid.1	$/\$ $/\$

**Assertion**
Ref	Expression
nnregexmid	$\/$

Distinct variable group:

Proof of Theorem nnregexmid Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3079	. . . 4 $\/$ $/\$
2		peano1 4335	. . . . 5
3		suc0 4166	. . . . . 6
4		peano2 4336	. . . . . . 7
5	2, 4	ax-mp 7	. . . . . 6
6	3, 5	eqeltrri 2152	. . . . 5
7		prssi 3543	. . . . 5 $/\$
8	2, 6, 7	mp2an 416	. . . 4
9	1, 8	sstri 3008	. . 3 $\/$ $/\$
10		eqid 2081	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4275	. . 3 $\/$ $/\$
12		pp0ex 3960	. . . . 5
13	12	rabex 3922	. . . 4 $\/$ $/\$
14		sseq1 3020	. . . . . 6 $\/$ $/\$ $\/$ $/\$
15		eleq2 2142	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
16	15	exbidv 1746	. . . . . 6 $\/$ $/\$ $\/$ $/\$
17	14, 16	anbi12d 456	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
18		eleq2 2142	. . . . . . . . . 10 $\/$ $/\$ $\/$ $/\$
19	18	notbid 624	. . . . . . . . 9 $\/$ $/\$ $\/$ $/\$
20	19	imbi2d 228	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$
21	20	albidv 1745	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
22	15, 21	anbi12d 456	. . . . . 6 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
23	22	exbidv 1746	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	17, 23	imbi12d 232	. . . 4 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
25		nnregexmid.1	. . . 4 $/\$ $/\$
26	13, 24, 25	vtocl 2653	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 416	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4276	. 2 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$
29	27, 28	ax-mp 7	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102 $\/$ wo 661

wal 1282

wceq 1284

wex 1421

wcel 1433

crab 2352

wss 2973

c0 3251

csn 3398

cpr 3399

csuc 4120

com 4331

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-suc 4126 df-iom 4332

This theorem is referenced by: (None)

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