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Theorem opthprc 4783

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 2295	. . . . 5
2		0ex 4221	. . . . . . . . 9
3	2	snid 3704	. . . . . . . 8
4		opelxp 4761	. . . . . . . 8 $/\$
5	3, 4	mpbiran2 950	. . . . . . 7
6		opelxp 4761	. . . . . . . 8 $/\$
7		0nep0 4261	. . . . . . . . . 10
8	2	elsn 3689	. . . . . . . . . 10
9	7, 8	nemtbir 2492	. . . . . . . . 9
10	9	bianfi 956	. . . . . . . 8 $/\$
11	6, 10	bitr4i 187	. . . . . . 7
12	5, 11	orbi12i 772	. . . . . 6 $\/$ $\/$
13		elun 3350	. . . . . 6 $\/$
14	9	biorfi 754	. . . . . 6 $\/$
15	12, 13, 14	3bitr4ri 213	. . . . 5
16		opelxp 4761	. . . . . . . 8 $/\$
17	3, 16	mpbiran2 950	. . . . . . 7
18		opelxp 4761	. . . . . . . 8 $/\$
19	9	bianfi 956	. . . . . . . 8 $/\$
20	18, 19	bitr4i 187	. . . . . . 7
21	17, 20	orbi12i 772	. . . . . 6 $\/$ $\/$
22		elun 3350	. . . . . 6 $\/$
23	9	biorfi 754	. . . . . 6 $\/$
24	21, 22, 23	3bitr4ri 213	. . . . 5
25	1, 15, 24	3bitr4g 223	. . . 4
26	25	eqrdv 2229	. . 3
27		eleq2 2295	. . . . 5
28		opelxp 4761	. . . . . . . 8 $/\$
29		p0ex 4284	. . . . . . . . . . . 12
30	29	elsn 3689	. . . . . . . . . . 11
31		eqcom 2233	. . . . . . . . . . 11
32	30, 31	bitri 184	. . . . . . . . . 10
33	7, 32	nemtbir 2492	. . . . . . . . 9
34	33	bianfi 956	. . . . . . . 8 $/\$
35	28, 34	bitr4i 187	. . . . . . 7
36	29	snid 3704	. . . . . . . 8
37		opelxp 4761	. . . . . . . 8 $/\$
38	36, 37	mpbiran2 950	. . . . . . 7
39	35, 38	orbi12i 772	. . . . . 6 $\/$ $\/$
40		elun 3350	. . . . . 6 $\/$
41		biorf 752	. . . . . . 7 $\/$
42	33, 41	ax-mp 5	. . . . . 6 $\/$
43	39, 40, 42	3bitr4ri 213	. . . . 5
44		opelxp 4761	. . . . . . . 8 $/\$
45	33	bianfi 956	. . . . . . . 8 $/\$
46	44, 45	bitr4i 187	. . . . . . 7
47		opelxp 4761	. . . . . . . 8 $/\$
48	36, 47	mpbiran2 950	. . . . . . 7
49	46, 48	orbi12i 772	. . . . . 6 $\/$ $\/$
50		elun 3350	. . . . . 6 $\/$
51		biorf 752	. . . . . . 7 $\/$
52	33, 51	ax-mp 5	. . . . . 6 $\/$
53	49, 50, 52	3bitr4ri 213	. . . . 5
54	27, 43, 53	3bitr4g 223	. . . 4
55	54	eqrdv 2229	. . 3
56	26, 55	jca 306	. 2 $/\$
57		xpeq1 4745	. . 3
58		xpeq1 4745	. . 3
59		uneq12 3358	. . 3 $/\$
60	57, 58, 59	syl2an 289	. 2 $/\$
61	56, 60	impbii 126	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104

wb 105 $\/$ wo 716

wceq 1398

wcel 2202

cun 3199

c0 3496

csn 3673

cop 3676

cxp 4729

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-v 2805 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-opab 4156 df-xp 4737

This theorem is referenced by: (None)

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