dfon2lem4 - Mathbox for Scott Fenton

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Theorem dfon2lem4 31691

Description: Lemma for dfon2 31697. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)

**Hypotheses**
Ref	Expression
dfon2lem4.1
dfon2lem4.2

**Assertion**
Ref	Expression
dfon2lem4	$/\$ $/\$ $/\$ $\/$

Distinct variable groups:

Proof of Theorem dfon2lem4 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3833	. . . . . . . . 9
2	1	sseli 3599	. . . . . . . 8
3		dfon2lem4.1	. . . . . . . . . . . 12
4		dfon2lem3 31690	. . . . . . . . . . . 12 $/\$ $/\$
5	3, 4	ax-mp 5	. . . . . . . . . . 11 $/\$ $/\$
6	5	simprd 479	. . . . . . . . . 10 $/\$
7		eleq1 2689	. . . . . . . . . . . . 13
8		eleq2 2690	. . . . . . . . . . . . 13
9	7, 8	bitrd 268	. . . . . . . . . . . 12
10	9	notbid 308	. . . . . . . . . . 11
11	10	rspccv 3306	. . . . . . . . . 10
12	6, 11	syl 17	. . . . . . . . 9 $/\$
13	12	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$
14	2, 13	syl5 34	. . . . . . 7 $/\$ $/\$ $/\$
15	14	pm2.01d 181	. . . . . 6 $/\$ $/\$ $/\$
16		elin 3796	. . . . . 6 $/\$
17	15, 16	sylnib 318	. . . . 5 $/\$ $/\$ $/\$ $/\$
18	5	simpld 475	. . . . . . . 8 $/\$
19		dfon2lem4.2	. . . . . . . . . 10
20		dfon2lem3 31690	. . . . . . . . . 10 $/\$ $/\$
21	19, 20	ax-mp 5	. . . . . . . . 9 $/\$ $/\$
22	21	simpld 475	. . . . . . . 8 $/\$
23		trin 4763	. . . . . . . 8 $/\$
24	18, 22, 23	syl2an 494	. . . . . . 7 $/\$ $/\$ $/\$
25	3	inex1 4799	. . . . . . . . 9
26		psseq1 3694	. . . . . . . . . . 11
27		treq 4758	. . . . . . . . . . 11
28	26, 27	anbi12d 747	. . . . . . . . . 10 $/\$ $/\$
29		eleq1 2689	. . . . . . . . . 10
30	28, 29	imbi12d 334	. . . . . . . . 9 $/\$ $/\$
31	25, 30	spcv 3299	. . . . . . . 8 $/\$ $/\$
32	31	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
33	24, 32	mpan2d 710	. . . . . 6 $/\$ $/\$ $/\$
34		psseq1 3694	. . . . . . . . . . 11
35		treq 4758	. . . . . . . . . . 11
36	34, 35	anbi12d 747	. . . . . . . . . 10 $/\$ $/\$
37		eleq1 2689	. . . . . . . . . 10
38	36, 37	imbi12d 334	. . . . . . . . 9 $/\$ $/\$
39	25, 38	spcv 3299	. . . . . . . 8 $/\$ $/\$
40	39	adantl 482	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	24, 40	mpan2d 710	. . . . . 6 $/\$ $/\$ $/\$
42	33, 41	anim12d 586	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
43	17, 42	mtod 189	. . . 4 $/\$ $/\$ $/\$ $/\$
44		ianor 509	. . . 4 $/\$ $\/$
45	43, 44	sylib 208	. . 3 $/\$ $/\$ $/\$ $\/$
46		sspss 3706	. . . . 5 $\/$
47	1, 46	mpbi 220	. . . 4 $\/$
48		inss2 3834	. . . . 5
49		sspss 3706	. . . . 5 $\/$
50	48, 49	mpbi 220	. . . 4 $\/$
51		orel1 397	. . . . . 6 $\/$
52		orc 400	. . . . . 6 $\/$
53	51, 52	syl6 35	. . . . 5 $\/$ $\/$
54		orel1 397	. . . . . 6 $\/$
55		olc 399	. . . . . 6 $\/$
56	54, 55	syl6 35	. . . . 5 $\/$ $\/$
57	53, 56	jaoa 532	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$
58	47, 50, 57	mp2ani 714	. . 3 $\/$ $\/$
59	45, 58	syl 17	. 2 $/\$ $/\$ $/\$ $\/$
60		df-ss 3588	. . 3
61		sseqin2 3817	. . 3
62	60, 61	orbi12i 543	. 2 $\/$ $\/$
63	59, 62	sylibr 224	1 $/\$ $/\$ $/\$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

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

wal 1481

wceq 1483

wcel 1990

wral 2912

cvv 3200

cin 3573

wss 3574

wpss 3575

wtr 4752

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-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949

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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-pw 4160 df-sn 4178 df-pr 4180 df-uni 4437 df-iun 4522 df-tr 4753 df-suc 5729

This theorem is referenced by: dfon2lem5 31692

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