tfr0dm - Intuitionistic Logic Explorer

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Theorem tfr0dm 6227

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 4063	. . . . 5
2		opexg 4158	. . . . 5 $/\$
3	1, 2	mpan 421	. . . 4
4		snidg 3561	. . . 4
5	3, 4	syl 14	. . 3
6		fnsng 5178	. . . . 5 $/\$
7	1, 6	mpan 421	. . . 4
8		fvsng 5624	. . . . . . 7 $/\$
9	1, 8	mpan 421	. . . . . 6
10		res0 4831	. . . . . . 7
11	10	fveq2i 5432	. . . . . 6
12	9, 11	eqtr4di 2191	. . . . 5
13		fveq2 5429	. . . . . . 7
14		reseq2 4822	. . . . . . . 8
15	14	fveq2d 5433	. . . . . . 7
16	13, 15	eqeq12d 2155	. . . . . 6
17	1, 16	ralsn 3574	. . . . 5
18	12, 17	sylibr 133	. . . 4
19		suc0 4341	. . . . . 6
20		0elon 4322	. . . . . . 7
21	20	onsuci 4440	. . . . . 6
22	19, 21	eqeltrri 2214	. . . . 5
23		fneq2 5220	. . . . . . 7
24		raleq 2629	. . . . . . 7
25	23, 24	anbi12d 465	. . . . . 6 $/\$ $/\$
26	25	rspcev 2793	. . . . 5 $/\$ $/\$ $/\$
27	22, 26	mpan 421	. . . 4 $/\$ $/\$
28	7, 18, 27	syl2anc 409	. . 3 $/\$
29		snexg 4116	. . . . 5
30		eleq2 2204	. . . . . . 7
31		fneq1 5219	. . . . . . . . 9
32		fveq1 5428	. . . . . . . . . . 11
33		reseq1 4821	. . . . . . . . . . . 12
34	33	fveq2d 5433	. . . . . . . . . . 11
35	32, 34	eqeq12d 2155	. . . . . . . . . 10
36	35	ralbidv 2438	. . . . . . . . 9
37	31, 36	anbi12d 465	. . . . . . . 8 $/\$ $/\$
38	37	rexbidv 2439	. . . . . . 7 $/\$ $/\$
39	30, 38	anbi12d 465	. . . . . 6 $/\$ $/\$ $/\$ $/\$
40	39	spcegv 2777	. . . . 5 $/\$ $/\$ $/\$ $/\$
41	3, 29, 40	3syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$
42		tfr.1	. . . . . 6 recs
43	42	eleq2i 2207	. . . . 5 recs
44		df-recs 6210	. . . . . 6 recs $/\$
45	44	eleq2i 2207	. . . . 5 recs $/\$
46		eluniab 3756	. . . . 5 $/\$ $/\$ $/\$
47	43, 45, 46	3bitri 205	. . . 4 $/\$ $/\$
48	41, 47	syl6ibr 161	. . 3 $/\$ $/\$
49	5, 28, 48	mp2and 430	. 2
50		opeldmg 4752	. . 3 $/\$
51	1, 50	mpan 421	. 2
52	49, 51	mpd 13	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1332

wex 1469

wcel 1481

cab 2126

wral 2417

wrex 2418

cvv 2689

c0 3368

csn 3532

cop 3535

cuni 3744

con0 4293

csuc 4295

cdm 4547

cres 4549

wfn 5126

cfv 5131 recscrecs 6209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-res 4559 df-iota 5096 df-fun 5133 df-fn 5134 df-fv 5139 df-recs 6210

This theorem is referenced by: tfr0 6228

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