nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4542

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4458 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6403 or nntri3or 6397), 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 3187	. . . 4 $\/$ $/\$
2		peano1 4516	. . . . 5
3		suc0 4341	. . . . . 6
4		peano2 4517	. . . . . . 7
5	2, 4	ax-mp 5	. . . . . 6
6	3, 5	eqeltrri 2214	. . . . 5
7		prssi 3686	. . . . 5 $/\$
8	2, 6, 7	mp2an 423	. . . 4
9	1, 8	sstri 3111	. . 3 $\/$ $/\$
10		eqid 2140	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4455	. . 3 $\/$ $/\$
12		pp0ex 4121	. . . . 5
13	12	rabex 4080	. . . 4 $\/$ $/\$
14		sseq1 3125	. . . . . 6 $\/$ $/\$ $\/$ $/\$
15		eleq2 2204	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
16	15	exbidv 1798	. . . . . 6 $\/$ $/\$ $\/$ $/\$
17	14, 16	anbi12d 465	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
18		eleq2 2204	. . . . . . . . . 10 $\/$ $/\$ $\/$ $/\$
19	18	notbid 657	. . . . . . . . 9 $\/$ $/\$ $\/$ $/\$
20	19	imbi2d 229	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$
21	20	albidv 1797	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
22	15, 21	anbi12d 465	. . . . . 6 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
23	22	exbidv 1798	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	17, 23	imbi12d 233	. . . 4 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
25		nnregexmid.1	. . . 4 $/\$ $/\$
26	13, 24, 25	vtocl 2743	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 423	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4456	. 2 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$
29	27, 28	ax-mp 5	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $\/$ wo 698

wal 1330

wceq 1332

wex 1469

wcel 1481

crab 2421

wss 3076

c0 3368

csn 3532

cpr 3533

csuc 4295

com 4512

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-suc 4301 df-iom 4513

This theorem is referenced by: (None)

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