nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4653

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4567 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6552 or nntri3or 6546), 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 3264	. . . 4 $\/$ $/\$
2		peano1 4626	. . . . 5
3		suc0 4442	. . . . . 6
4		peano2 4627	. . . . . . 7
5	2, 4	ax-mp 5	. . . . . 6
6	3, 5	eqeltrri 2267	. . . . 5
7		prssi 3776	. . . . 5 $/\$
8	2, 6, 7	mp2an 426	. . . 4
9	1, 8	sstri 3188	. . 3 $\/$ $/\$
10		eqid 2193	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4564	. . 3 $\/$ $/\$
12		pp0ex 4218	. . . . 5
13	12	rabex 4173	. . . 4 $\/$ $/\$
14		sseq1 3202	. . . . . 6 $\/$ $/\$ $\/$ $/\$
15		eleq2 2257	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
16	15	exbidv 1836	. . . . . 6 $\/$ $/\$ $\/$ $/\$
17	14, 16	anbi12d 473	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
18		eleq2 2257	. . . . . . . . . 10 $\/$ $/\$ $\/$ $/\$
19	18	notbid 668	. . . . . . . . 9 $\/$ $/\$ $\/$ $/\$
20	19	imbi2d 230	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$
21	20	albidv 1835	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
22	15, 21	anbi12d 473	. . . . . 6 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
23	22	exbidv 1836	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	17, 23	imbi12d 234	. . . 4 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
25		nnregexmid.1	. . . 4 $/\$ $/\$
26	13, 24, 25	vtocl 2814	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 426	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4565	. 2 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$
29	27, 28	ax-mp 5	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 709

wal 1362

wceq 1364

wex 1503

wcel 2164

crab 2476

wss 3153

c0 3446

csn 3618

cpr 3619

csuc 4396

com 4622

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-uni 3836 df-int 3871 df-suc 4402 df-iom 4623

This theorem is referenced by: (None)

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