frec0g - Intuitionistic Logic Explorer

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

Theorem frec0g 6006

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 4567	. . . . . . . . . 10
2	1	biantrur 297	. . . . . . . . 9 $/\$
3		vex 2604	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4173	. . . . . . . . . . . . . . . 16
5	3, 4	ax-mp 7	. . . . . . . . . . . . . . 15
6	5	nesymi 2291	. . . . . . . . . . . . . 14
7	1	eqeq1i 2088	. . . . . . . . . . . . . 14
8	6, 7	mtbir 628	. . . . . . . . . . . . 13
9	8	intnanr 872	. . . . . . . . . . . 12 $/\$
10	9	a1i 9	. . . . . . . . . . 11 $/\$
11	10	nrex 2453	. . . . . . . . . 10 $/\$
12	11	biorfi 697	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$
13		orcom 679	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
14	2, 12, 13	3bitri 204	. . . . . . . 8 $/\$ $\/$ $/\$
15	14	abbii 2194	. . . . . . 7 $/\$ $\/$ $/\$
16		abid2 2199	. . . . . . 7
17	15, 16	eqtr3i 2103	. . . . . 6 $/\$ $\/$ $/\$
18		elex 2610	. . . . . 6
19	17, 18	syl5eqel 2165	. . . . 5 $/\$ $\/$ $/\$
20		0ex 3905	. . . . . . 7
21		dmeq 4553	. . . . . . . . . . . . 13
22	21	eqeq1d 2089	. . . . . . . . . . . 12
23		fveq1 5197	. . . . . . . . . . . . . 14
24	23	fveq2d 5202	. . . . . . . . . . . . 13
25	24	eleq2d 2148	. . . . . . . . . . . 12
26	22, 25	anbi12d 456	. . . . . . . . . . 11 $/\$ $/\$
27	26	rexbidv 2369	. . . . . . . . . 10 $/\$ $/\$
28	21	eqeq1d 2089	. . . . . . . . . . 11
29	28	anbi1d 452	. . . . . . . . . 10 $/\$ $/\$
30	27, 29	orbi12d 739	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
31	30	abbidv 2196	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
32		eqid 2081	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
33	31, 32	fvmptg 5269	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
34	20, 33	mpan 414	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
35	34, 17	syl6eq 2129	. . . . 5 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
36	19, 35	syl 14	. . . 4 $/\$ $\/$ $/\$
37	36, 18	eqeltrd 2155	. . 3 $/\$ $\/$ $/\$
38		df-frec 6001	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5199	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4335	. . . . . 6
41		fvres 5219	. . . . . 6 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
42	40, 41	ax-mp 7	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
43	39, 42	eqtri 2101	. . . 4 frec recs $/\$ $\/$ $/\$
44		eqid 2081	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
45	44	tfr0 5960	. . . 4 $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
46	43, 45	syl5eq 2125	. . 3 $/\$ $\/$ $/\$ frec $/\$ $\/$ $/\$
47	37, 46	syl 14	. 2 frec $/\$ $\/$ $/\$
48	47, 36	eqtrd 2113	1 frec

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 102 $\/$ wo 661

wceq 1284

wcel 1433

cab 2067

wne 2245

wrex 2349

cvv 2601

c0 3251

cmpt 3839

csuc 4120

com 4331

cdm 4363

cres 4365

cfv 4922 recscrecs 5942 freccfrec 6000

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-res 4375 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 df-recs 5943 df-frec 6001

This theorem is referenced by: frecrdg 6015 freccl 6016 frec2uz0d 9401 frec2uzrdg 9411 frecuzrdg0 9416

W3C validator