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Theorem opthprc 4518

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 2158	. . . . 5
2		0ex 3987	. . . . . . . . 9
3	2	snid 3495	. . . . . . . 8
4		opelxp 4497	. . . . . . . 8 $/\$
5	3, 4	mpbiran2 890	. . . . . . 7
6		opelxp 4497	. . . . . . . 8 $/\$
7		0nep0 4021	. . . . . . . . . 10
8	2	elsn 3482	. . . . . . . . . 10
9	7, 8	nemtbir 2351	. . . . . . . . 9
10	9	bianfi 896	. . . . . . . 8 $/\$
11	6, 10	bitr4i 186	. . . . . . 7
12	5, 11	orbi12i 719	. . . . . 6 $\/$ $\/$
13		elun 3156	. . . . . 6 $\/$
14	9	biorfi 703	. . . . . 6 $\/$
15	12, 13, 14	3bitr4ri 212	. . . . 5
16		opelxp 4497	. . . . . . . 8 $/\$
17	3, 16	mpbiran2 890	. . . . . . 7
18		opelxp 4497	. . . . . . . 8 $/\$
19	9	bianfi 896	. . . . . . . 8 $/\$
20	18, 19	bitr4i 186	. . . . . . 7
21	17, 20	orbi12i 719	. . . . . 6 $\/$ $\/$
22		elun 3156	. . . . . 6 $\/$
23	9	biorfi 703	. . . . . 6 $\/$
24	21, 22, 23	3bitr4ri 212	. . . . 5
25	1, 15, 24	3bitr4g 222	. . . 4
26	25	eqrdv 2093	. . 3
27		eleq2 2158	. . . . 5
28		opelxp 4497	. . . . . . . 8 $/\$
29		p0ex 4044	. . . . . . . . . . . 12
30	29	elsn 3482	. . . . . . . . . . 11
31		eqcom 2097	. . . . . . . . . . 11
32	30, 31	bitri 183	. . . . . . . . . 10
33	7, 32	nemtbir 2351	. . . . . . . . 9
34	33	bianfi 896	. . . . . . . 8 $/\$
35	28, 34	bitr4i 186	. . . . . . 7
36	29	snid 3495	. . . . . . . 8
37		opelxp 4497	. . . . . . . 8 $/\$
38	36, 37	mpbiran2 890	. . . . . . 7
39	35, 38	orbi12i 719	. . . . . 6 $\/$ $\/$
40		elun 3156	. . . . . 6 $\/$
41		biorf 701	. . . . . . 7 $\/$
42	33, 41	ax-mp 7	. . . . . 6 $\/$
43	39, 40, 42	3bitr4ri 212	. . . . 5
44		opelxp 4497	. . . . . . . 8 $/\$
45	33	bianfi 896	. . . . . . . 8 $/\$
46	44, 45	bitr4i 186	. . . . . . 7
47		opelxp 4497	. . . . . . . 8 $/\$
48	36, 47	mpbiran2 890	. . . . . . 7
49	46, 48	orbi12i 719	. . . . . 6 $\/$ $\/$
50		elun 3156	. . . . . 6 $\/$
51		biorf 701	. . . . . . 7 $\/$
52	33, 51	ax-mp 7	. . . . . 6 $\/$
53	49, 50, 52	3bitr4ri 212	. . . . 5
54	27, 43, 53	3bitr4g 222	. . . 4
55	54	eqrdv 2093	. . 3
56	26, 55	jca 301	. 2 $/\$
57		xpeq1 4481	. . 3
58		xpeq1 4481	. . 3
59		uneq12 3164	. . 3 $/\$
60	57, 58, 59	syl2an 284	. 2 $/\$
61	56, 60	impbii 125	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103

wb 104 $\/$ wo 667

wceq 1296

wcel 1445

cun 3011

c0 3302

csn 3466

cop 3469

cxp 4465

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060

This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-opab 3922 df-xp 4473

This theorem is referenced by: (None)

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