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Theorem opthprc 4409

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 2142	. . . . 5
2		0ex 3905	. . . . . . . . 9
3	2	snid 3425	. . . . . . . 8
4		opelxp 4392	. . . . . . . 8 $/\$
5	3, 4	mpbiran2 882	. . . . . . 7
6		opelxp 4392	. . . . . . . 8 $/\$
7		0nep0 3939	. . . . . . . . . 10
8	2	elsn 3414	. . . . . . . . . 10
9	7, 8	nemtbir 2334	. . . . . . . . 9
10	9	bianfi 888	. . . . . . . 8 $/\$
11	6, 10	bitr4i 185	. . . . . . 7
12	5, 11	orbi12i 713	. . . . . 6 $\/$ $\/$
13		elun 3113	. . . . . 6 $\/$
14	9	biorfi 697	. . . . . 6 $\/$
15	12, 13, 14	3bitr4ri 211	. . . . 5
16		opelxp 4392	. . . . . . . 8 $/\$
17	3, 16	mpbiran2 882	. . . . . . 7
18		opelxp 4392	. . . . . . . 8 $/\$
19	9	bianfi 888	. . . . . . . 8 $/\$
20	18, 19	bitr4i 185	. . . . . . 7
21	17, 20	orbi12i 713	. . . . . 6 $\/$ $\/$
22		elun 3113	. . . . . 6 $\/$
23	9	biorfi 697	. . . . . 6 $\/$
24	21, 22, 23	3bitr4ri 211	. . . . 5
25	1, 15, 24	3bitr4g 221	. . . 4
26	25	eqrdv 2079	. . 3
27		eleq2 2142	. . . . 5
28		opelxp 4392	. . . . . . . 8 $/\$
29		p0ex 3959	. . . . . . . . . . . 12
30	29	elsn 3414	. . . . . . . . . . 11
31		eqcom 2083	. . . . . . . . . . 11
32	30, 31	bitri 182	. . . . . . . . . 10
33	7, 32	nemtbir 2334	. . . . . . . . 9
34	33	bianfi 888	. . . . . . . 8 $/\$
35	28, 34	bitr4i 185	. . . . . . 7
36	29	snid 3425	. . . . . . . 8
37		opelxp 4392	. . . . . . . 8 $/\$
38	36, 37	mpbiran2 882	. . . . . . 7
39	35, 38	orbi12i 713	. . . . . 6 $\/$ $\/$
40		elun 3113	. . . . . 6 $\/$
41		biorf 695	. . . . . . 7 $\/$
42	33, 41	ax-mp 7	. . . . . 6 $\/$
43	39, 40, 42	3bitr4ri 211	. . . . 5
44		opelxp 4392	. . . . . . . 8 $/\$
45	33	bianfi 888	. . . . . . . 8 $/\$
46	44, 45	bitr4i 185	. . . . . . 7
47		opelxp 4392	. . . . . . . 8 $/\$
48	36, 47	mpbiran2 882	. . . . . . 7
49	46, 48	orbi12i 713	. . . . . 6 $\/$ $\/$
50		elun 3113	. . . . . 6 $\/$
51		biorf 695	. . . . . . 7 $\/$
52	33, 51	ax-mp 7	. . . . . 6 $\/$
53	49, 50, 52	3bitr4ri 211	. . . . 5
54	27, 43, 53	3bitr4g 221	. . . 4
55	54	eqrdv 2079	. . 3
56	26, 55	jca 300	. 2 $/\$
57		xpeq1 4377	. . 3
58		xpeq1 4377	. . 3
59		uneq12 3121	. . 3 $/\$
60	57, 58, 59	syl2an 283	. 2 $/\$
61	56, 60	impbii 124	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 102

wb 103 $\/$ wo 661

wceq 1284

wcel 1433

cun 2971

c0 3251

csn 3398

cop 3401

cxp 4361

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 df-xp 4369

This theorem is referenced by: (None)

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