nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4529

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4445 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6388 or nntri3or 6382), 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 3177	. . . 4 $\/$ $/\$
2		peano1 4503	. . . . 5
3		suc0 4328	. . . . . 6
4		peano2 4504	. . . . . . 7
5	2, 4	ax-mp 5	. . . . . 6
6	3, 5	eqeltrri 2211	. . . . 5
7		prssi 3673	. . . . 5 $/\$
8	2, 6, 7	mp2an 422	. . . 4
9	1, 8	sstri 3101	. . 3 $\/$ $/\$
10		eqid 2137	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4442	. . 3 $\/$ $/\$
12		pp0ex 4108	. . . . 5
13	12	rabex 4067	. . . 4 $\/$ $/\$
14		sseq1 3115	. . . . . 6 $\/$ $/\$ $\/$ $/\$
15		eleq2 2201	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
16	15	exbidv 1797	. . . . . 6 $\/$ $/\$ $\/$ $/\$
17	14, 16	anbi12d 464	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
18		eleq2 2201	. . . . . . . . . 10 $\/$ $/\$ $\/$ $/\$
19	18	notbid 656	. . . . . . . . 9 $\/$ $/\$ $\/$ $/\$
20	19	imbi2d 229	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$
21	20	albidv 1796	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
22	15, 21	anbi12d 464	. . . . . 6 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
23	22	exbidv 1797	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	17, 23	imbi12d 233	. . . 4 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
25		nnregexmid.1	. . . 4 $/\$ $/\$
26	13, 24, 25	vtocl 2735	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 422	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4443	. 2 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$
29	27, 28	ax-mp 5	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $\/$ wo 697

wal 1329

wceq 1331

wex 1468

wcel 1480

crab 2418

wss 3066

c0 3358

csn 3522

cpr 3523

csuc 4282

com 4499

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-suc 4288 df-iom 4500

This theorem is referenced by: (None)

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