nnregexmid - Intuitionistic Logic Explorer

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Theorem nnregexmid 4632

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4546 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6513 or nntri3or 6507), 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 3252	. . . 4 $\/$ $/\$
2		peano1 4605	. . . . 5
3		suc0 4423	. . . . . 6
4		peano2 4606	. . . . . . 7
5	2, 4	ax-mp 5	. . . . . 6
6	3, 5	eqeltrri 2261	. . . . 5
7		prssi 3762	. . . . 5 $/\$
8	2, 6, 7	mp2an 426	. . . 4
9	1, 8	sstri 3176	. . 3 $\/$ $/\$
10		eqid 2187	. . . 4 $\/$ $/\$ $\/$ $/\$
11	10	regexmidlemm 4543	. . 3 $\/$ $/\$
12		pp0ex 4201	. . . . 5
13	12	rabex 4159	. . . 4 $\/$ $/\$
14		sseq1 3190	. . . . . 6 $\/$ $/\$ $\/$ $/\$
15		eleq2 2251	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
16	15	exbidv 1835	. . . . . 6 $\/$ $/\$ $\/$ $/\$
17	14, 16	anbi12d 473	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
18		eleq2 2251	. . . . . . . . . 10 $\/$ $/\$ $\/$ $/\$
19	18	notbid 668	. . . . . . . . 9 $\/$ $/\$ $\/$ $/\$
20	19	imbi2d 230	. . . . . . . 8 $\/$ $/\$ $\/$ $/\$
21	20	albidv 1834	. . . . . . 7 $\/$ $/\$ $\/$ $/\$
22	15, 21	anbi12d 473	. . . . . 6 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
23	22	exbidv 1835	. . . . 5 $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	17, 23	imbi12d 234	. . . 4 $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
25		nnregexmid.1	. . . 4 $/\$ $/\$
26	13, 24, 25	vtocl 2803	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	9, 11, 26	mp2an 426	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
28	10	regexmidlem1 4544	. 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 1361

wceq 1363

wex 1502

wcel 2158

crab 2469

wss 3141

c0 3434

csn 3604

cpr 3605

csuc 4377

com 4601

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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445

This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-uni 3822 df-int 3857 df-suc 4383 df-iom 4602

This theorem is referenced by: (None)

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