r1sdom - Metamath Proof Explorer

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Theorem r1sdom 8637

Description: Each stage in the cumulative hierarchy is strictly larger than the last. (Contributed by Mario Carneiro, 19-Apr-2013.)

**Assertion**
Ref	Expression
r1sdom	$/\$

Proof of Theorem r1sdom Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2690	. . . 4
2		fveq2 6191	. . . . 5
3	2	breq2d 4665	. . . 4
4	1, 3	imbi12d 334	. . 3
5		eleq2 2690	. . . 4
6		fveq2 6191	. . . . 5
7	6	breq2d 4665	. . . 4
8	5, 7	imbi12d 334	. . 3
9		eleq2 2690	. . . 4
10		fveq2 6191	. . . . 5
11	10	breq2d 4665	. . . 4
12	9, 11	imbi12d 334	. . 3
13		eleq2 2690	. . . 4
14		fveq2 6191	. . . . 5
15	14	breq2d 4665	. . . 4
16	13, 15	imbi12d 334	. . 3
17		noel 3919	. . . 4
18	17	pm2.21i 116	. . 3
19		elsuci 5791	. . . . 5 $\/$
20		sdomtr 8098	. . . . . . . . 9 $/\$
21	20	expcom 451	. . . . . . . 8
22		fvex 6201	. . . . . . . . . 10
23	22	canth2 8113	. . . . . . . . 9
24		r1suc 8633	. . . . . . . . 9
25	23, 24	syl5breqr 4691	. . . . . . . 8
26	21, 25	syl11 33	. . . . . . 7
27	26	imim2i 16	. . . . . 6
28		fveq2 6191	. . . . . . . . 9
29	28	breq1d 4663	. . . . . . . 8
30	25, 29	syl5ibr 236	. . . . . . 7
31	30	a1i 11	. . . . . 6
32	27, 31	jaod 395	. . . . 5 $\/$
33	19, 32	syl5 34	. . . 4
34	33	com3r 87	. . 3
35		limuni 5785	. . . . . . 7
36	35	eleq2d 2687	. . . . . 6
37		eluni2 4440	. . . . . 6
38	36, 37	syl6bb 276	. . . . 5
39		r19.29 3072	. . . . . . 7 $/\$ $/\$
40		fvex 6201	. . . . . . . . . . 11
41	40	a1i 11	. . . . . . . . . 10
42		ssiun2 4563	. . . . . . . . . . 11
43		vex 3203	. . . . . . . . . . . . 13
44		r1lim 8635	. . . . . . . . . . . . 13 $/\$
45	43, 44	mpan 706	. . . . . . . . . . . 12
46	45	sseq2d 3633	. . . . . . . . . . 11
47	42, 46	syl5ibr 236	. . . . . . . . . 10
48		ssdomg 8001	. . . . . . . . . 10
49	41, 47, 48	sylsyld 61	. . . . . . . . 9
50		id 22	. . . . . . . . . . 11
51	50	imp 445	. . . . . . . . . 10 $/\$
52		sdomdomtr 8093	. . . . . . . . . . 11 $/\$
53	52	expcom 451	. . . . . . . . . 10
54	51, 53	syl5 34	. . . . . . . . 9 $/\$
55	49, 54	syl6 35	. . . . . . . 8 $/\$
56	55	rexlimdv 3030	. . . . . . 7 $/\$
57	39, 56	syl5 34	. . . . . 6 $/\$
58	57	expcomd 454	. . . . 5
59	38, 58	sylbid 230	. . . 4
60	59	com23 86	. . 3
61	4, 8, 12, 16, 18, 34, 60	tfinds 7059	. 2
62	61	imp 445	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

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

wceq 1483

wcel 1990

wral 2912

wrex 2913

cvv 3200

wss 3574

c0 3915

cpw 4158

cuni 4436

ciun 4520 class class class wbr 4653

con0 5723

wlim 5724

csuc 5725

cfv 5888

cdom 7953

csdm 7954

cr1 8625

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

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-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-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-r1 8627

This theorem is referenced by: r111 8638 smobeth 9408 r1tskina 9604

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