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Theorem opthprc 4655

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 2230	. . . . 5
2		0ex 4109	. . . . . . . . 9
3	2	snid 3607	. . . . . . . 8
4		opelxp 4634	. . . . . . . 8 $/\$
5	3, 4	mpbiran2 931	. . . . . . 7
6		opelxp 4634	. . . . . . . 8 $/\$
7		0nep0 4144	. . . . . . . . . 10
8	2	elsn 3592	. . . . . . . . . 10
9	7, 8	nemtbir 2425	. . . . . . . . 9
10	9	bianfi 937	. . . . . . . 8 $/\$
11	6, 10	bitr4i 186	. . . . . . 7
12	5, 11	orbi12i 754	. . . . . 6 $\/$ $\/$
13		elun 3263	. . . . . 6 $\/$
14	9	biorfi 736	. . . . . 6 $\/$
15	12, 13, 14	3bitr4ri 212	. . . . 5
16		opelxp 4634	. . . . . . . 8 $/\$
17	3, 16	mpbiran2 931	. . . . . . 7
18		opelxp 4634	. . . . . . . 8 $/\$
19	9	bianfi 937	. . . . . . . 8 $/\$
20	18, 19	bitr4i 186	. . . . . . 7
21	17, 20	orbi12i 754	. . . . . 6 $\/$ $\/$
22		elun 3263	. . . . . 6 $\/$
23	9	biorfi 736	. . . . . 6 $\/$
24	21, 22, 23	3bitr4ri 212	. . . . 5
25	1, 15, 24	3bitr4g 222	. . . 4
26	25	eqrdv 2163	. . 3
27		eleq2 2230	. . . . 5
28		opelxp 4634	. . . . . . . 8 $/\$
29		p0ex 4167	. . . . . . . . . . . 12
30	29	elsn 3592	. . . . . . . . . . 11
31		eqcom 2167	. . . . . . . . . . 11
32	30, 31	bitri 183	. . . . . . . . . 10
33	7, 32	nemtbir 2425	. . . . . . . . 9
34	33	bianfi 937	. . . . . . . 8 $/\$
35	28, 34	bitr4i 186	. . . . . . 7
36	29	snid 3607	. . . . . . . 8
37		opelxp 4634	. . . . . . . 8 $/\$
38	36, 37	mpbiran2 931	. . . . . . 7
39	35, 38	orbi12i 754	. . . . . 6 $\/$ $\/$
40		elun 3263	. . . . . 6 $\/$
41		biorf 734	. . . . . . 7 $\/$
42	33, 41	ax-mp 5	. . . . . 6 $\/$
43	39, 40, 42	3bitr4ri 212	. . . . 5
44		opelxp 4634	. . . . . . . 8 $/\$
45	33	bianfi 937	. . . . . . . 8 $/\$
46	44, 45	bitr4i 186	. . . . . . 7
47		opelxp 4634	. . . . . . . 8 $/\$
48	36, 47	mpbiran2 931	. . . . . . 7
49	46, 48	orbi12i 754	. . . . . 6 $\/$ $\/$
50		elun 3263	. . . . . 6 $\/$
51		biorf 734	. . . . . . 7 $\/$
52	33, 51	ax-mp 5	. . . . . 6 $\/$
53	49, 50, 52	3bitr4ri 212	. . . . 5
54	27, 43, 53	3bitr4g 222	. . . 4
55	54	eqrdv 2163	. . 3
56	26, 55	jca 304	. 2 $/\$
57		xpeq1 4618	. . 3
58		xpeq1 4618	. . 3
59		uneq12 3271	. . 3 $/\$
60	57, 58, 59	syl2an 287	. 2 $/\$
61	56, 60	impbii 125	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103

wb 104 $\/$ wo 698

wceq 1343

wcel 2136

cun 3114

c0 3409

csn 3576

cop 3579

cxp 4602

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-v 2728 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-opab 4044 df-xp 4610

This theorem is referenced by: (None)

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