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Theorem opthprc 4726

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 2269	. . . . 5
2		0ex 4171	. . . . . . . . 9
3	2	snid 3664	. . . . . . . 8
4		opelxp 4705	. . . . . . . 8 $/\$
5	3, 4	mpbiran2 944	. . . . . . 7
6		opelxp 4705	. . . . . . . 8 $/\$
7		0nep0 4209	. . . . . . . . . 10
8	2	elsn 3649	. . . . . . . . . 10
9	7, 8	nemtbir 2465	. . . . . . . . 9
10	9	bianfi 950	. . . . . . . 8 $/\$
11	6, 10	bitr4i 187	. . . . . . 7
12	5, 11	orbi12i 766	. . . . . 6 $\/$ $\/$
13		elun 3314	. . . . . 6 $\/$
14	9	biorfi 748	. . . . . 6 $\/$
15	12, 13, 14	3bitr4ri 213	. . . . 5
16		opelxp 4705	. . . . . . . 8 $/\$
17	3, 16	mpbiran2 944	. . . . . . 7
18		opelxp 4705	. . . . . . . 8 $/\$
19	9	bianfi 950	. . . . . . . 8 $/\$
20	18, 19	bitr4i 187	. . . . . . 7
21	17, 20	orbi12i 766	. . . . . 6 $\/$ $\/$
22		elun 3314	. . . . . 6 $\/$
23	9	biorfi 748	. . . . . 6 $\/$
24	21, 22, 23	3bitr4ri 213	. . . . 5
25	1, 15, 24	3bitr4g 223	. . . 4
26	25	eqrdv 2203	. . 3
27		eleq2 2269	. . . . 5
28		opelxp 4705	. . . . . . . 8 $/\$
29		p0ex 4232	. . . . . . . . . . . 12
30	29	elsn 3649	. . . . . . . . . . 11
31		eqcom 2207	. . . . . . . . . . 11
32	30, 31	bitri 184	. . . . . . . . . 10
33	7, 32	nemtbir 2465	. . . . . . . . 9
34	33	bianfi 950	. . . . . . . 8 $/\$
35	28, 34	bitr4i 187	. . . . . . 7
36	29	snid 3664	. . . . . . . 8
37		opelxp 4705	. . . . . . . 8 $/\$
38	36, 37	mpbiran2 944	. . . . . . 7
39	35, 38	orbi12i 766	. . . . . 6 $\/$ $\/$
40		elun 3314	. . . . . 6 $\/$
41		biorf 746	. . . . . . 7 $\/$
42	33, 41	ax-mp 5	. . . . . 6 $\/$
43	39, 40, 42	3bitr4ri 213	. . . . 5
44		opelxp 4705	. . . . . . . 8 $/\$
45	33	bianfi 950	. . . . . . . 8 $/\$
46	44, 45	bitr4i 187	. . . . . . 7
47		opelxp 4705	. . . . . . . 8 $/\$
48	36, 47	mpbiran2 944	. . . . . . 7
49	46, 48	orbi12i 766	. . . . . 6 $\/$ $\/$
50		elun 3314	. . . . . 6 $\/$
51		biorf 746	. . . . . . 7 $\/$
52	33, 51	ax-mp 5	. . . . . 6 $\/$
53	49, 50, 52	3bitr4ri 213	. . . . 5
54	27, 43, 53	3bitr4g 223	. . . 4
55	54	eqrdv 2203	. . 3
56	26, 55	jca 306	. 2 $/\$
57		xpeq1 4689	. . 3
58		xpeq1 4689	. . 3
59		uneq12 3322	. . 3 $/\$
60	57, 58, 59	syl2an 289	. 2 $/\$
61	56, 60	impbii 126	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104

wb 105 $\/$ wo 710

wceq 1373

wcel 2176

cun 3164

c0 3460

csn 3633

cop 3636

cxp 4673

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-ral 2489 df-rex 2490 df-v 2774 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-opab 4106 df-xp 4681

This theorem is referenced by: (None)

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