frec0g - Intuitionistic Logic Explorer

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

Theorem frec0g 6606

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 4951	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2806	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4521	. . . . . . . . . . . . . . . 16
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15
6	5	nesymi 2449	. . . . . . . . . . . . . 14
7	1	eqeq1i 2239	. . . . . . . . . . . . . 14
8	6, 7	mtbir 678	. . . . . . . . . . . . 13
9	8	intnanr 938	. . . . . . . . . . . 12 $/\$
10	9	a1i 9	. . . . . . . . . . 11 $/\$
11	10	nrex 2625	. . . . . . . . . 10 $/\$
12	11	biorfi 754	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$
13		orcom 736	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
14	2, 12, 13	3bitri 206	. . . . . . . 8 $/\$ $\/$ $/\$
15	14	abbii 2347	. . . . . . 7 $/\$ $\/$ $/\$
16		abid2 2353	. . . . . . 7
17	15, 16	eqtr3i 2254	. . . . . 6 $/\$ $\/$ $/\$
18		elex 2815	. . . . . 6
19	17, 18	eqeltrid 2318	. . . . 5 $/\$ $\/$ $/\$
20		0ex 4221	. . . . . . 7
21		dmeq 4937	. . . . . . . . . . . . 13
22	21	eqeq1d 2240	. . . . . . . . . . . 12
23		fveq1 5647	. . . . . . . . . . . . . 14
24	23	fveq2d 5652	. . . . . . . . . . . . 13
25	24	eleq2d 2301	. . . . . . . . . . . 12
26	22, 25	anbi12d 473	. . . . . . . . . . 11 $/\$ $/\$
27	26	rexbidv 2534	. . . . . . . . . 10 $/\$ $/\$
28	21	eqeq1d 2240	. . . . . . . . . . 11
29	28	anbi1d 465	. . . . . . . . . 10 $/\$ $/\$
30	27, 29	orbi12d 801	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
31	30	abbidv 2350	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
32		eqid 2231	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
33	31, 32	fvmptg 5731	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
34	20, 33	mpan 424	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
35	34, 17	eqtrdi 2280	. . . . 5 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
36	19, 35	syl 14	. . . 4 $/\$ $\/$ $/\$
37	36, 18	eqeltrd 2308	. . 3 $/\$ $\/$ $/\$
38		df-frec 6600	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5649	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4698	. . . . . 6
41		fvres 5672	. . . . . 6 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
42	40, 41	ax-mp 5	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
43	39, 42	eqtri 2252	. . . 4 frec recs $/\$ $\/$ $/\$
44		eqid 2231	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
45	44	tfr0 6532	. . . 4 $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
46	43, 45	eqtrid 2276	. . 3 $/\$ $\/$ $/\$ frec $/\$ $\/$ $/\$
47	37, 46	syl 14	. 2 frec $/\$ $\/$ $/\$
48	47, 36	eqtrd 2264	1 frec

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 716

wceq 1398

wcel 2202

cab 2217

wne 2403

wrex 2512

cvv 2803

c0 3496

cmpt 4155

csuc 4468

com 4694

cdm 4731

cres 4733

cfv 5333 recscrecs 6513 freccfrec 6599

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-res 4743 df-iota 5293 df-fun 5335 df-fn 5336 df-fv 5341 df-recs 6514 df-frec 6600

This theorem is referenced by: frecrdg 6617 frec2uz0d 10724 frec2uzrdg 10734 frecuzrdg0 10738 frecuzrdgg 10741 frecuzrdg0t 10747 seq3val 10785 seqvalcd 10786

W3C validator