nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4598

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4512 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6467 or nntri3or 6461), 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 3227	. . . 4 $\/$ $/\$
2		peano1 4571	. . . . 5
3		suc0 4389	. . . . . 6
4		peano2 4572	. . . . . . 7
5	2, 4	ax-mp 5	. . . . . 6
6	3, 5	eqeltrri 2240	. . . . 5
7		prssi 3731	. . . . 5 $/\$
8	2, 6, 7	mp2an 423	. . . 4
9	1, 8	sstri 3151	. . 3 $\/$ $/\$
10		eqid 2165	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4509	. . 3 $\/$ $/\$
12		pp0ex 4168	. . . . 5
13	12	rabex 4126	. . . 4 $\/$ $/\$
14		sseq1 3165	. . . . . 6 $\/$ $/\$ $\/$ $/\$
15		eleq2 2230	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
16	15	exbidv 1813	. . . . . 6 $\/$ $/\$ $\/$ $/\$
17	14, 16	anbi12d 465	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
18		eleq2 2230	. . . . . . . . . 10 $\/$ $/\$ $\/$ $/\$
19	18	notbid 657	. . . . . . . . 9 $\/$ $/\$ $\/$ $/\$
20	19	imbi2d 229	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$
21	20	albidv 1812	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
22	15, 21	anbi12d 465	. . . . . 6 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
23	22	exbidv 1813	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	17, 23	imbi12d 233	. . . 4 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
25		nnregexmid.1	. . . 4 $/\$ $/\$
26	13, 24, 25	vtocl 2780	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 423	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4510	. 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 1341

wceq 1343

wex 1480

wcel 2136

crab 2448

wss 3116

c0 3409

csn 3576

cpr 3577

csuc 4343

com 4567

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-uni 3790 df-int 3825 df-suc 4349 df-iom 4568

This theorem is referenced by: (None)

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