sbthlem2 - Metamath Proof Explorer

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Theorem sbthlem2 8071

Description: Lemma for sbth 8080. (Contributed by NM, 22-Mar-1998.)

**Hypotheses**
Ref	Expression
sbthlem.1
sbthlem.2	$/\$ $\$ $\$

**Assertion**
Ref	Expression
sbthlem2	$\$ $\$

Distinct variable groups:

Allowed substitution hints:

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**Proof of Theorem sbthlem2**
Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . . . . 9
2		sbthlem.2	. . . . . . . . 9 $/\$ $\$ $\$
3	1, 2	sbthlem1 8070	. . . . . . . 8 $\$ $\$
4		imass2 5501	. . . . . . . 8 $\$ $\$ $\$ $\$
5		sscon 3744	. . . . . . . 8 $\$ $\$ $\$ $\$ $\$ $\$
6	3, 4, 5	mp2b 10	. . . . . . 7 $\$ $\$ $\$ $\$
7		imass2 5501	. . . . . . 7 $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$
8		sscon 3744	. . . . . . 7 $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$
9	6, 7, 8	mp2b 10	. . . . . 6 $\$ $\$ $\$ $\$ $\$ $\$
10		imassrn 5477	. . . . . . . 8 $\$ $\$ $\$
11		sstr2 3610	. . . . . . . 8 $\$ $\$ $\$ $\$ $\$ $\$
12	10, 11	ax-mp 5	. . . . . . 7 $\$ $\$ $\$
13		difss 3737	. . . . . . 7 $\$ $\$
14		ssconb 3743	. . . . . . 7 $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$
15	12, 13, 14	sylancl 694	. . . . . 6 $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$
16	9, 15	mpbiri 248	. . . . 5 $\$ $\$ $\$ $\$ $\$ $\$
17	16, 13	jctil 560	. . . 4 $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
18	1, 13	ssexi 4803	. . . . 5 $\$ $\$
19		sseq1 3626	. . . . . 6 $\$ $\$ $\$ $\$
20		imaeq2 5462	. . . . . . . . 9 $\$ $\$ $\$ $\$
21	20	difeq2d 3728	. . . . . . . 8 $\$ $\$ $\$ $\$ $\$ $\$
22	21	imaeq2d 5466	. . . . . . 7 $\$ $\$ $\$ $\$ $\$ $\$
23		difeq2 3722	. . . . . . 7 $\$ $\$ $\$ $\$ $\$ $\$
24	22, 23	sseq12d 3634	. . . . . 6 $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$
25	19, 24	anbi12d 747	. . . . 5 $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
26	18, 25	elab 3350	. . . 4 $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$
27	17, 26	sylibr 224	. . 3 $\$ $\$ $/\$ $\$ $\$
28	27, 2	syl6eleqr 2712	. 2 $\$ $\$
29		elssuni 4467	. 2 $\$ $\$ $\$ $\$
30	28, 29	syl 17	1 $\$ $\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

cab 2608

cvv 3200 $\$ cdif 3571

wss 3574

cuni 4436

crn 5115

cima 5117

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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127

This theorem is referenced by: sbthlem3 8072

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