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Theorem opthprc 4598

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 2204	. . . . 5
2		0ex 4063	. . . . . . . . 9
3	2	snid 3563	. . . . . . . 8
4		opelxp 4577	. . . . . . . 8 $/\$
5	3, 4	mpbiran2 926	. . . . . . 7
6		opelxp 4577	. . . . . . . 8 $/\$
7		0nep0 4097	. . . . . . . . . 10
8	2	elsn 3548	. . . . . . . . . 10
9	7, 8	nemtbir 2398	. . . . . . . . 9
10	9	bianfi 932	. . . . . . . 8 $/\$
11	6, 10	bitr4i 186	. . . . . . 7
12	5, 11	orbi12i 754	. . . . . 6 $\/$ $\/$
13		elun 3222	. . . . . 6 $\/$
14	9	biorfi 736	. . . . . 6 $\/$
15	12, 13, 14	3bitr4ri 212	. . . . 5
16		opelxp 4577	. . . . . . . 8 $/\$
17	3, 16	mpbiran2 926	. . . . . . 7
18		opelxp 4577	. . . . . . . 8 $/\$
19	9	bianfi 932	. . . . . . . 8 $/\$
20	18, 19	bitr4i 186	. . . . . . 7
21	17, 20	orbi12i 754	. . . . . 6 $\/$ $\/$
22		elun 3222	. . . . . 6 $\/$
23	9	biorfi 736	. . . . . 6 $\/$
24	21, 22, 23	3bitr4ri 212	. . . . 5
25	1, 15, 24	3bitr4g 222	. . . 4
26	25	eqrdv 2138	. . 3
27		eleq2 2204	. . . . 5
28		opelxp 4577	. . . . . . . 8 $/\$
29		p0ex 4120	. . . . . . . . . . . 12
30	29	elsn 3548	. . . . . . . . . . 11
31		eqcom 2142	. . . . . . . . . . 11
32	30, 31	bitri 183	. . . . . . . . . 10
33	7, 32	nemtbir 2398	. . . . . . . . 9
34	33	bianfi 932	. . . . . . . 8 $/\$
35	28, 34	bitr4i 186	. . . . . . 7
36	29	snid 3563	. . . . . . . 8
37		opelxp 4577	. . . . . . . 8 $/\$
38	36, 37	mpbiran2 926	. . . . . . 7
39	35, 38	orbi12i 754	. . . . . 6 $\/$ $\/$
40		elun 3222	. . . . . 6 $\/$
41		biorf 734	. . . . . . 7 $\/$
42	33, 41	ax-mp 5	. . . . . 6 $\/$
43	39, 40, 42	3bitr4ri 212	. . . . 5
44		opelxp 4577	. . . . . . . 8 $/\$
45	33	bianfi 932	. . . . . . . 8 $/\$
46	44, 45	bitr4i 186	. . . . . . 7
47		opelxp 4577	. . . . . . . 8 $/\$
48	36, 47	mpbiran2 926	. . . . . . 7
49	46, 48	orbi12i 754	. . . . . 6 $\/$ $\/$
50		elun 3222	. . . . . 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 2138	. . 3
56	26, 55	jca 304	. 2 $/\$
57		xpeq1 4561	. . 3
58		xpeq1 4561	. . 3
59		uneq12 3230	. . 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 1332

wcel 1481

cun 3074

c0 3368

csn 3532

cop 3535

cxp 4545

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 df-xp 4553

This theorem is referenced by: (None)

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