frec0g - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > frec0g		Unicode version

Theorem frec0g 5983

Description: The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)

**Assertion**
Ref	Expression
frec0g	frec

Proof of Theorem frec0g Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dm0 4549	. . . . . . . . . 10
2	1	biantrur 287	. . . . . . . . 9 $/\$
3		vex 2560	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4155	. . . . . . . . . . . . . . . 16
5	3, 4	ax-mp 7	. . . . . . . . . . . . . . 15
6	5	nesymi 2251	. . . . . . . . . . . . . 14
7	1	eqeq1i 2047	. . . . . . . . . . . . . 14
8	6, 7	mtbir 596	. . . . . . . . . . . . 13
9	8	intnanr 839	. . . . . . . . . . . 12 $/\$
10	9	a1i 9	. . . . . . . . . . 11 $/\$
11	10	nrex 2411	. . . . . . . . . 10 $/\$
12	11	biorfi 665	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$
13		orcom 647	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
14	2, 12, 13	3bitri 195	. . . . . . . 8 $/\$ $\/$ $/\$
15	14	abbii 2153	. . . . . . 7 $/\$ $\/$ $/\$
16		abid2 2158	. . . . . . 7
17	15, 16	eqtr3i 2062	. . . . . 6 $/\$ $\/$ $/\$
18		elex 2566	. . . . . 6
19	17, 18	syl5eqel 2124	. . . . 5 $/\$ $\/$ $/\$
20		0ex 3884	. . . . . . 7
21		dmeq 4535	. . . . . . . . . . . . 13
22	21	eqeq1d 2048	. . . . . . . . . . . 12
23		fveq1 5177	. . . . . . . . . . . . . 14
24	23	fveq2d 5182	. . . . . . . . . . . . 13
25	24	eleq2d 2107	. . . . . . . . . . . 12
26	22, 25	anbi12d 442	. . . . . . . . . . 11 $/\$ $/\$
27	26	rexbidv 2327	. . . . . . . . . 10 $/\$ $/\$
28	21	eqeq1d 2048	. . . . . . . . . . 11
29	28	anbi1d 438	. . . . . . . . . 10 $/\$ $/\$
30	27, 29	orbi12d 707	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
31	30	abbidv 2155	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
32		eqid 2040	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
33	31, 32	fvmptg 5248	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
34	20, 33	mpan 400	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
35	34, 17	syl6eq 2088	. . . . 5 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
36	19, 35	syl 14	. . . 4 $/\$ $\/$ $/\$
37	36, 18	eqeltrd 2114	. . 3 $/\$ $\/$ $/\$
38		df-frec 5978	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5179	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4317	. . . . . 6
41		fvres 5198	. . . . . 6 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
42	40, 41	ax-mp 7	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
43	39, 42	eqtri 2060	. . . 4 frec recs $/\$ $\/$ $/\$
44		eqid 2040	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
45	44	tfr0 5937	. . . 4 $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
46	43, 45	syl5eq 2084	. . 3 $/\$ $\/$ $/\$ frec $/\$ $\/$ $/\$
47	37, 46	syl 14	. 2 frec $/\$ $\/$ $/\$
48	47, 36	eqtrd 2072	1 frec

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 97 $\/$ wo 629

wceq 1243

wcel 1393

cab 2026

wne 2204

wrex 2307

cvv 2557

c0 3224

cmpt 3818

csuc 4102

com 4313

cdm 4345

cres 4347

cfv 4902 recscrecs 5919 freccfrec 5977

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-res 4357 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 df-recs 5920 df-frec 5978

This theorem is referenced by: frecrdg 5992 freccl 5993 frec2uz0d 9185 frec2uzrdg 9195 frecuzrdg0 9200

W3C validator