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Theorem opthprc 4662

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 2234	. . . . 5
2		0ex 4116	. . . . . . . . 9
3	2	snid 3614	. . . . . . . 8
4		opelxp 4641	. . . . . . . 8 $/\$
5	3, 4	mpbiran2 936	. . . . . . 7
6		opelxp 4641	. . . . . . . 8 $/\$
7		0nep0 4151	. . . . . . . . . 10
8	2	elsn 3599	. . . . . . . . . 10
9	7, 8	nemtbir 2429	. . . . . . . . 9
10	9	bianfi 942	. . . . . . . 8 $/\$
11	6, 10	bitr4i 186	. . . . . . 7
12	5, 11	orbi12i 759	. . . . . 6 $\/$ $\/$
13		elun 3268	. . . . . 6 $\/$
14	9	biorfi 741	. . . . . 6 $\/$
15	12, 13, 14	3bitr4ri 212	. . . . 5
16		opelxp 4641	. . . . . . . 8 $/\$
17	3, 16	mpbiran2 936	. . . . . . 7
18		opelxp 4641	. . . . . . . 8 $/\$
19	9	bianfi 942	. . . . . . . 8 $/\$
20	18, 19	bitr4i 186	. . . . . . 7
21	17, 20	orbi12i 759	. . . . . 6 $\/$ $\/$
22		elun 3268	. . . . . 6 $\/$
23	9	biorfi 741	. . . . . 6 $\/$
24	21, 22, 23	3bitr4ri 212	. . . . 5
25	1, 15, 24	3bitr4g 222	. . . 4
26	25	eqrdv 2168	. . 3
27		eleq2 2234	. . . . 5
28		opelxp 4641	. . . . . . . 8 $/\$
29		p0ex 4174	. . . . . . . . . . . 12
30	29	elsn 3599	. . . . . . . . . . 11
31		eqcom 2172	. . . . . . . . . . 11
32	30, 31	bitri 183	. . . . . . . . . 10
33	7, 32	nemtbir 2429	. . . . . . . . 9
34	33	bianfi 942	. . . . . . . 8 $/\$
35	28, 34	bitr4i 186	. . . . . . 7
36	29	snid 3614	. . . . . . . 8
37		opelxp 4641	. . . . . . . 8 $/\$
38	36, 37	mpbiran2 936	. . . . . . 7
39	35, 38	orbi12i 759	. . . . . 6 $\/$ $\/$
40		elun 3268	. . . . . 6 $\/$
41		biorf 739	. . . . . . 7 $\/$
42	33, 41	ax-mp 5	. . . . . 6 $\/$
43	39, 40, 42	3bitr4ri 212	. . . . 5
44		opelxp 4641	. . . . . . . 8 $/\$
45	33	bianfi 942	. . . . . . . 8 $/\$
46	44, 45	bitr4i 186	. . . . . . 7
47		opelxp 4641	. . . . . . . 8 $/\$
48	36, 47	mpbiran2 936	. . . . . . 7
49	46, 48	orbi12i 759	. . . . . 6 $\/$ $\/$
50		elun 3268	. . . . . 6 $\/$
51		biorf 739	. . . . . . 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 2168	. . 3
56	26, 55	jca 304	. 2 $/\$
57		xpeq1 4625	. . 3
58		xpeq1 4625	. . 3
59		uneq12 3276	. . 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 703

wceq 1348

wcel 2141

cun 3119

c0 3414

csn 3583

cop 3586

cxp 4609

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-opab 4051 df-xp 4617

This theorem is referenced by: (None)

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