tfr0dm - Intuitionistic Logic Explorer

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Theorem tfr0dm 6408

Description: Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.)

**Hypothesis**
Ref	Expression
tfr.1	recs

**Assertion**
Ref	Expression
tfr0dm

Proof of Theorem tfr0dm Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4171	. . . . 5
2		opexg 4272	. . . . 5 $/\$
3	1, 2	mpan 424	. . . 4
4		snidg 3662	. . . 4
5	3, 4	syl 14	. . 3
6		fnsng 5321	. . . . 5 $/\$
7	1, 6	mpan 424	. . . 4
8		fvsng 5780	. . . . . . 7 $/\$
9	1, 8	mpan 424	. . . . . 6
10		res0 4963	. . . . . . 7
11	10	fveq2i 5579	. . . . . 6
12	9, 11	eqtr4di 2256	. . . . 5
13		fveq2 5576	. . . . . . 7
14		reseq2 4954	. . . . . . . 8
15	14	fveq2d 5580	. . . . . . 7
16	13, 15	eqeq12d 2220	. . . . . 6
17	1, 16	ralsn 3676	. . . . 5
18	12, 17	sylibr 134	. . . 4
19		suc0 4458	. . . . . 6
20		0elon 4439	. . . . . . 7
21	20	onsuci 4564	. . . . . 6
22	19, 21	eqeltrri 2279	. . . . 5
23		fneq2 5363	. . . . . . 7
24		raleq 2702	. . . . . . 7
25	23, 24	anbi12d 473	. . . . . 6 $/\$ $/\$
26	25	rspcev 2877	. . . . 5 $/\$ $/\$ $/\$
27	22, 26	mpan 424	. . . 4 $/\$ $/\$
28	7, 18, 27	syl2anc 411	. . 3 $/\$
29		snexg 4228	. . . . 5
30		eleq2 2269	. . . . . . 7
31		fneq1 5362	. . . . . . . . 9
32		fveq1 5575	. . . . . . . . . . 11
33		reseq1 4953	. . . . . . . . . . . 12
34	33	fveq2d 5580	. . . . . . . . . . 11
35	32, 34	eqeq12d 2220	. . . . . . . . . 10
36	35	ralbidv 2506	. . . . . . . . 9
37	31, 36	anbi12d 473	. . . . . . . 8 $/\$ $/\$
38	37	rexbidv 2507	. . . . . . 7 $/\$ $/\$
39	30, 38	anbi12d 473	. . . . . 6 $/\$ $/\$ $/\$ $/\$
40	39	spcegv 2861	. . . . 5 $/\$ $/\$ $/\$ $/\$
41	3, 29, 40	3syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$
42		tfr.1	. . . . . 6 recs
43	42	eleq2i 2272	. . . . 5 recs
44		df-recs 6391	. . . . . 6 recs $/\$
45	44	eleq2i 2272	. . . . 5 recs $/\$
46		eluniab 3862	. . . . 5 $/\$ $/\$ $/\$
47	43, 45, 46	3bitri 206	. . . 4 $/\$ $/\$
48	41, 47	imbitrrdi 162	. . 3 $/\$ $/\$
49	5, 28, 48	mp2and 433	. 2
50		opeldmg 4883	. . 3 $/\$
51	1, 50	mpan 424	. 2
52	49, 51	mpd 13	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1373

wex 1515

wcel 2176

cab 2191

wral 2484

wrex 2485

cvv 2772

c0 3460

csn 3633

cop 3636

cuni 3850

con0 4410

csuc 4412

cdm 4675

cres 4677

wfn 5266

cfv 5271 recscrecs 6390

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-tr 4143 df-id 4340 df-iord 4413 df-on 4415 df-suc 4418 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-res 4687 df-iota 5232 df-fun 5273 df-fn 5274 df-fv 5279 df-recs 6391

This theorem is referenced by: tfr0 6409

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