nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4654

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4568 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6554 or nntri3or 6548), 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 3265	. . . 4 $\/$ $/\$
2		peano1 4627	. . . . 5
3		suc0 4443	. . . . . 6
4		peano2 4628	. . . . . . 7
5	2, 4	ax-mp 5	. . . . . 6
6	3, 5	eqeltrri 2267	. . . . 5
7		prssi 3777	. . . . 5 $/\$
8	2, 6, 7	mp2an 426	. . . 4
9	1, 8	sstri 3189	. . 3 $\/$ $/\$
10		eqid 2193	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4565	. . 3 $\/$ $/\$
12		pp0ex 4219	. . . . 5
13	12	rabex 4174	. . . 4 $\/$ $/\$
14		sseq1 3203	. . . . . 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 2815	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 426	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4566	. 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 3154

c0 3447

csn 3619

cpr 3620

csuc 4397

com 4623

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 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465

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 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-uni 3837 df-int 3872 df-suc 4403 df-iom 4624

This theorem is referenced by: (None)

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