nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4447

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4364 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6274 or nntri3or 6268), 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 3107	. . . 4 $\/$ $/\$
2		peano1 4422	. . . . 5
3		suc0 4247	. . . . . 6
4		peano2 4423	. . . . . . 7
5	2, 4	ax-mp 7	. . . . . 6
6	3, 5	eqeltrri 2162	. . . . 5
7		prssi 3601	. . . . 5 $/\$
8	2, 6, 7	mp2an 418	. . . 4
9	1, 8	sstri 3035	. . 3 $\/$ $/\$
10		eqid 2089	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4361	. . 3 $\/$ $/\$
12		pp0ex 4030	. . . . 5
13	12	rabex 3989	. . . 4 $\/$ $/\$
14		sseq1 3048	. . . . . 6 $\/$ $/\$ $\/$ $/\$
15		eleq2 2152	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
16	15	exbidv 1754	. . . . . 6 $\/$ $/\$ $\/$ $/\$
17	14, 16	anbi12d 458	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
18		eleq2 2152	. . . . . . . . . 10 $\/$ $/\$ $\/$ $/\$
19	18	notbid 628	. . . . . . . . 9 $\/$ $/\$ $\/$ $/\$
20	19	imbi2d 229	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$
21	20	albidv 1753	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
22	15, 21	anbi12d 458	. . . . . 6 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
23	22	exbidv 1754	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	17, 23	imbi12d 233	. . . 4 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
25		nnregexmid.1	. . . 4 $/\$ $/\$
26	13, 24, 25	vtocl 2674	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 418	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4362	. 2 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$
29	27, 28	ax-mp 7	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $\/$ wo 665

wal 1288

wceq 1290

wex 1427

wcel 1439

crab 2364

wss 3000

c0 3287

csn 3450

cpr 3451

csuc 4201

com 4418

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269

This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-uni 3660 df-int 3695 df-suc 4207 df-iom 4419

This theorem is referenced by: (None)

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