nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4621

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4535 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6500 or nntri3or 6494), 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 3241	. . . 4 $\/$ $/\$
2		peano1 4594	. . . . 5
3		suc0 4412	. . . . . 6
4		peano2 4595	. . . . . . 7
5	2, 4	ax-mp 5	. . . . . 6
6	3, 5	eqeltrri 2251	. . . . 5
7		prssi 3751	. . . . 5 $/\$
8	2, 6, 7	mp2an 426	. . . 4
9	1, 8	sstri 3165	. . 3 $\/$ $/\$
10		eqid 2177	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4532	. . 3 $\/$ $/\$
12		pp0ex 4190	. . . . 5
13	12	rabex 4148	. . . 4 $\/$ $/\$
14		sseq1 3179	. . . . . 6 $\/$ $/\$ $\/$ $/\$
15		eleq2 2241	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
16	15	exbidv 1825	. . . . . 6 $\/$ $/\$ $\/$ $/\$
17	14, 16	anbi12d 473	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
18		eleq2 2241	. . . . . . . . . 10 $\/$ $/\$ $\/$ $/\$
19	18	notbid 667	. . . . . . . . 9 $\/$ $/\$ $\/$ $/\$
20	19	imbi2d 230	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$
21	20	albidv 1824	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
22	15, 21	anbi12d 473	. . . . . 6 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
23	22	exbidv 1825	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	17, 23	imbi12d 234	. . . 4 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
25		nnregexmid.1	. . . 4 $/\$ $/\$
26	13, 24, 25	vtocl 2792	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 426	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4533	. 2 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$
29	27, 28	ax-mp 5	1 $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 708

wal 1351

wceq 1353

wex 1492

wcel 2148

crab 2459

wss 3130

c0 3423

csn 3593

cpr 3594

csuc 4366

com 4590

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-uni 3811 df-int 3846 df-suc 4372 df-iom 4591

This theorem is referenced by: (None)

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