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Theorem opthprc 5167

Description: Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.)

**Assertion**
Ref	Expression
opthprc	$/\$

Proof of Theorem opthprc Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2690	. . . . 5
2		0ex 4790	. . . . . . . . 9
3	2	snid 4208	. . . . . . . 8
4		opelxp 5146	. . . . . . . 8 $/\$
5	3, 4	mpbiran2 954	. . . . . . 7
6		opelxp 5146	. . . . . . . 8 $/\$
7		0nep0 4836	. . . . . . . . . 10
8	2	elsn 4192	. . . . . . . . . 10
9	7, 8	nemtbir 2889	. . . . . . . . 9
10	9	bianfi 966	. . . . . . . 8 $/\$
11	6, 10	bitr4i 267	. . . . . . 7
12	5, 11	orbi12i 543	. . . . . 6 $\/$ $\/$
13		elun 3753	. . . . . 6 $\/$
14	9	biorfi 422	. . . . . 6 $\/$
15	12, 13, 14	3bitr4ri 293	. . . . 5
16		opelxp 5146	. . . . . . . 8 $/\$
17	3, 16	mpbiran2 954	. . . . . . 7
18		opelxp 5146	. . . . . . . 8 $/\$
19	9	bianfi 966	. . . . . . . 8 $/\$
20	18, 19	bitr4i 267	. . . . . . 7
21	17, 20	orbi12i 543	. . . . . 6 $\/$ $\/$
22		elun 3753	. . . . . 6 $\/$
23	9	biorfi 422	. . . . . 6 $\/$
24	21, 22, 23	3bitr4ri 293	. . . . 5
25	1, 15, 24	3bitr4g 303	. . . 4
26	25	eqrdv 2620	. . 3
27		eleq2 2690	. . . . 5
28		opelxp 5146	. . . . . . . 8 $/\$
29		p0ex 4853	. . . . . . . . . . . 12
30	29	elsn 4192	. . . . . . . . . . 11
31		eqcom 2629	. . . . . . . . . . 11
32	30, 31	bitri 264	. . . . . . . . . 10
33	7, 32	nemtbir 2889	. . . . . . . . 9
34	33	bianfi 966	. . . . . . . 8 $/\$
35	28, 34	bitr4i 267	. . . . . . 7
36	29	snid 4208	. . . . . . . 8
37		opelxp 5146	. . . . . . . 8 $/\$
38	36, 37	mpbiran2 954	. . . . . . 7
39	35, 38	orbi12i 543	. . . . . 6 $\/$ $\/$
40		elun 3753	. . . . . 6 $\/$
41		biorf 420	. . . . . . 7 $\/$
42	33, 41	ax-mp 5	. . . . . 6 $\/$
43	39, 40, 42	3bitr4ri 293	. . . . 5
44		opelxp 5146	. . . . . . . 8 $/\$
45	33	bianfi 966	. . . . . . . 8 $/\$
46	44, 45	bitr4i 267	. . . . . . 7
47		opelxp 5146	. . . . . . . 8 $/\$
48	36, 47	mpbiran2 954	. . . . . . 7
49	46, 48	orbi12i 543	. . . . . 6 $\/$ $\/$
50		elun 3753	. . . . . 6 $\/$
51		biorf 420	. . . . . . 7 $\/$
52	33, 51	ax-mp 5	. . . . . 6 $\/$
53	49, 50, 52	3bitr4ri 293	. . . . 5
54	27, 43, 53	3bitr4g 303	. . . 4
55	54	eqrdv 2620	. . 3
56	26, 55	jca 554	. 2 $/\$
57		xpeq1 5128	. . 3
58		xpeq1 5128	. . 3
59		uneq12 3762	. . 3 $/\$
60	57, 58, 59	syl2an 494	. 2 $/\$
61	56, 60	impbii 199	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

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

wceq 1483

wcel 1990

cun 3572

c0 3915

csn 4177

cop 4183

cxp 5112

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-pow 4843 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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 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-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120

This theorem is referenced by: (None)

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