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Theorem rankxpsuc 8745

Description: The rank of a Cartesian product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 8742 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.)

**Hypotheses**
Ref	Expression
rankxplim.1
rankxplim.2

**Assertion**
Ref	Expression
rankxpsuc	$/\$

Proof of Theorem rankxpsuc Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rankuni 8726	. . . . . . . 8
2		rankuni 8726	. . . . . . . . 9
3	2	unieqi 4445	. . . . . . . 8
4	1, 3	eqtri 2644	. . . . . . 7
5		unixp 5668	. . . . . . . 8
6	5	fveq2d 6195	. . . . . . 7
7	4, 6	syl5reqr 2671	. . . . . 6
8		suc11reg 8516	. . . . . 6
9	7, 8	sylibr 224	. . . . 5
10	9	adantl 482	. . . 4 $/\$
11		fvex 6201	. . . . . . . . . . . . . 14
12		eleq1 2689	. . . . . . . . . . . . . 14
13	11, 12	mpbii 223	. . . . . . . . . . . . 13
14		sucexb 7009	. . . . . . . . . . . . 13
15	13, 14	sylibr 224	. . . . . . . . . . . 12
16		nlimsucg 7042	. . . . . . . . . . . 12
17	15, 16	syl 17	. . . . . . . . . . 11
18		limeq 5735	. . . . . . . . . . 11
19	17, 18	mtbird 315	. . . . . . . . . 10
20		rankxplim.1	. . . . . . . . . . 11
21		rankxplim.2	. . . . . . . . . . 11
22	20, 21	rankxplim2 8743	. . . . . . . . . 10
23	19, 22	nsyl 135	. . . . . . . . 9
24	20, 21	xpex 6962	. . . . . . . . . . . . . 14
25	24	rankeq0 8724	. . . . . . . . . . . . 13
26	25	necon3abii 2840	. . . . . . . . . . . 12
27		rankon 8658	. . . . . . . . . . . . . . . 16
28	27	onordi 5832	. . . . . . . . . . . . . . 15
29		ordzsl 7045	. . . . . . . . . . . . . . 15 $\/$ $\/$
30	28, 29	mpbi 220	. . . . . . . . . . . . . 14 $\/$ $\/$
31		3orass 1040	. . . . . . . . . . . . . 14 $\/$ $\/$ $\/$ $\/$
32	30, 31	mpbi 220	. . . . . . . . . . . . 13 $\/$ $\/$
33	32	ori 390	. . . . . . . . . . . 12 $\/$
34	26, 33	sylbi 207	. . . . . . . . . . 11 $\/$
35	34	ord 392	. . . . . . . . . 10
36	35	con1d 139	. . . . . . . . 9
37	23, 36	syl5com 31	. . . . . . . 8
38		vex 3203	. . . . . . . . . . . 12
39		nlimsucg 7042	. . . . . . . . . . . 12
40	38, 39	ax-mp 5	. . . . . . . . . . 11
41		limeq 5735	. . . . . . . . . . 11
42	40, 41	mtbiri 317	. . . . . . . . . 10
43	42	rexlimivw 3029	. . . . . . . . 9
44	20, 21	rankxplim3 8744	. . . . . . . . 9
45	43, 44	sylnib 318	. . . . . . . 8
46	37, 45	syl6com 37	. . . . . . 7
47		unixp0 5669	. . . . . . . . . . . 12
48	24	uniex 6953	. . . . . . . . . . . . 13
49	48	rankeq0 8724	. . . . . . . . . . . 12
50	2	eqeq1i 2627	. . . . . . . . . . . 12
51	47, 49, 50	3bitri 286	. . . . . . . . . . 11
52	51	necon3abii 2840	. . . . . . . . . 10
53		onuni 6993	. . . . . . . . . . . . . . 15
54	27, 53	ax-mp 5	. . . . . . . . . . . . . 14
55	54	onordi 5832	. . . . . . . . . . . . 13
56		ordzsl 7045	. . . . . . . . . . . . 13 $\/$ $\/$
57	55, 56	mpbi 220	. . . . . . . . . . . 12 $\/$ $\/$
58		3orass 1040	. . . . . . . . . . . 12 $\/$ $\/$ $\/$ $\/$
59	57, 58	mpbi 220	. . . . . . . . . . 11 $\/$ $\/$
60	59	ori 390	. . . . . . . . . 10 $\/$
61	52, 60	sylbi 207	. . . . . . . . 9 $\/$
62	61	ord 392	. . . . . . . 8
63	62	con1d 139	. . . . . . 7
64	46, 63	syld 47	. . . . . 6
65	64	impcom 446	. . . . 5 $/\$
66		onsucuni2 7034	. . . . . . 7 $/\$
67	54, 66	mpan 706	. . . . . 6
68	67	rexlimivw 3029	. . . . 5
69	65, 68	syl 17	. . . 4 $/\$
70	10, 69	eqtrd 2656	. . 3 $/\$
71		suc11reg 8516	. . 3
72	70, 71	sylibr 224	. 2 $/\$
73	37	imp 445	. . 3 $/\$
74		onsucuni2 7034	. . . . 5 $/\$
75	27, 74	mpan 706	. . . 4
76	75	rexlimivw 3029	. . 3
77	73, 76	syl 17	. 2 $/\$
78	72, 77	eqtr2d 2657	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $\/$ wo 383 $/\$ wa 384 $\/$ w3o 1036

wceq 1483

wcel 1990

wne 2794

wrex 2913

cvv 3200

cun 3572

c0 3915

cuni 4436

cxp 5112

word 5722

con0 5723

wlim 5724

csuc 5725

cfv 5888

crnk 8626

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 ax-inf2 8538

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-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-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-r1 8627 df-rank 8628

This theorem is referenced by: (None)

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