finxpsuclem - Mathbox for ML

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Theorem finxpsuclem 33234

Description: Lemma for finxpsuc 33235. (Contributed by ML, 24-Oct-2020.)

**Hypothesis**
Ref	Expression
finxpsuclem.1	$/\$

**Assertion**
Ref	Expression
finxpsuclem	$/\$

Distinct variable groups:

Allowed substitution hints:

(

)

Proof of Theorem finxpsuclem Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7086	. . . . . . . . . 10
2	1	adantr 481	. . . . . . . . 9 $/\$
3		1on 7567	. . . . . . . . . . . . 13
4	3	onordi 5832	. . . . . . . . . . . 12
5		nnord 7073	. . . . . . . . . . . 12
6		ordsseleq 5752	. . . . . . . . . . . 12 $/\$ $\/$
7	4, 5, 6	sylancr 695	. . . . . . . . . . 11 $\/$
8	7	biimpa 501	. . . . . . . . . 10 $/\$ $\/$
9		elelsuc 5797	. . . . . . . . . . . . 13
10	9	a1i 11	. . . . . . . . . . . 12
11		sucidg 5803	. . . . . . . . . . . . 13
12		eleq1 2689	. . . . . . . . . . . . 13
13	11, 12	syl5ibrcom 237	. . . . . . . . . . . 12
14	10, 13	jaod 395	. . . . . . . . . . 11 $\/$
15	14	adantr 481	. . . . . . . . . 10 $/\$ $\/$
16	8, 15	mpd 15	. . . . . . . . 9 $/\$
17		finxpsuclem.1	. . . . . . . . . 10 $/\$
18	17	finxpreclem6 33233	. . . . . . . . 9 $/\$
19	2, 16, 18	syl2anc 693	. . . . . . . 8 $/\$
20	19	sselda 3603	. . . . . . 7 $/\$ $/\$
21	1	ad2antrr 762	. . . . . . . . . . . . 13 $/\$ $/\$
22		df-2o 7561	. . . . . . . . . . . . . . 15
23		ordsucsssuc 7023	. . . . . . . . . . . . . . . . 17 $/\$
24	4, 5, 23	sylancr 695	. . . . . . . . . . . . . . . 16
25	24	biimpa 501	. . . . . . . . . . . . . . 15 $/\$
26	22, 25	syl5eqss 3649	. . . . . . . . . . . . . 14 $/\$
27	26	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$
28		simpr 477	. . . . . . . . . . . . 13 $/\$ $/\$
29	17	finxpreclem4 33231	. . . . . . . . . . . . 13 $/\$ $/\$
30	21, 27, 28, 29	syl21anc 1325	. . . . . . . . . . . 12 $/\$ $/\$
31		ordunisuc 7032	. . . . . . . . . . . . . . . 16
32	5, 31	syl 17	. . . . . . . . . . . . . . 15
33		opeq1 4402	. . . . . . . . . . . . . . . 16
34		rdgeq2 7508	. . . . . . . . . . . . . . . 16
35	33, 34	syl 17	. . . . . . . . . . . . . . 15
36	32, 35	syl 17	. . . . . . . . . . . . . 14
37	36, 32	fveq12d 6197	. . . . . . . . . . . . 13
38	37	ad2antrr 762	. . . . . . . . . . . 12 $/\$ $/\$
39	30, 38	eqtrd 2656	. . . . . . . . . . 11 $/\$ $/\$
40	39	eqeq2d 2632	. . . . . . . . . 10 $/\$ $/\$
41	1	biantrurd 529	. . . . . . . . . . . 12 $/\$
42	17	dffinxpf 33222	. . . . . . . . . . . . 13 $/\$
43	42	abeq2i 2735	. . . . . . . . . . . 12 $/\$
44	41, 43	syl6rbbr 279	. . . . . . . . . . 11
45	44	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$
46		fvex 6201	. . . . . . . . . . . . 13
47		opeq2 4403	. . . . . . . . . . . . . . . . 17
48		rdgeq2 7508	. . . . . . . . . . . . . . . . 17
49	47, 48	syl 17	. . . . . . . . . . . . . . . 16
50	49	fveq1d 6193	. . . . . . . . . . . . . . 15
51	50	eqeq2d 2632	. . . . . . . . . . . . . 14
52	51	anbi2d 740	. . . . . . . . . . . . 13 $/\$ $/\$
53	17	dffinxpf 33222	. . . . . . . . . . . . 13 $/\$
54	46, 52, 53	elab2 3354	. . . . . . . . . . . 12 $/\$
55	54	baib 944	. . . . . . . . . . 11
56	55	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$
57	40, 45, 56	3bitr4d 300	. . . . . . . . 9 $/\$ $/\$
58	57	biimpd 219	. . . . . . . 8 $/\$ $/\$
59	58	impancom 456	. . . . . . 7 $/\$ $/\$
60	20, 59	mpd 15	. . . . . 6 $/\$ $/\$
61	60	ex 450	. . . . 5 $/\$
62	20	ex 450	. . . . 5 $/\$
63	61, 62	jcad 555	. . . 4 $/\$ $/\$
64	57	exbiri 652	. . . . . 6 $/\$
65	64	impd 447	. . . . 5 $/\$ $/\$
66	65	ancomsd 470	. . . 4 $/\$ $/\$
67	63, 66	impbid 202	. . 3 $/\$ $/\$
68		elxp8 33219	. . 3 $/\$
69	67, 68	syl6bbr 278	. 2 $/\$
70	69	eqrdv 2620	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wceq 1483

wcel 1990

cvv 3200

wss 3574

c0 3915

cif 4086

cop 4183

cuni 4436

cxp 5112

word 5722

csuc 5725

cfv 5888

cmpt2 6652

com 7065

c1st 7166

crdg 7505

c1o 7553

c2o 7554

cfinxp 33220

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-finxp 33221

This theorem is referenced by: finxpsuc 33235

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