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Theorem opthprc 4821

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 2302	. . . . 5
2		0ex 4255	. . . . . . . . 9
3	2	snid 3736	. . . . . . . 8
4		opelxp 4799	. . . . . . . 8 $/\$
5	3, 4	mpbiran2 954	. . . . . . 7
6		opelxp 4799	. . . . . . . 8 $/\$
7		0nep0 4297	. . . . . . . . . 10
8	2	elsn 3721	. . . . . . . . . 10
9	7, 8	nemtbir 2509	. . . . . . . . 9
10	9	bianfi 960	. . . . . . . 8 $/\$
11	6, 10	bitr4i 187	. . . . . . 7
12	5, 11	orbi12i 776	. . . . . 6 $\/$ $\/$
13		elun 3370	. . . . . 6 $\/$
14	9	biorfi 758	. . . . . 6 $\/$
15	12, 13, 14	3bitr4ri 213	. . . . 5
16		opelxp 4799	. . . . . . . 8 $/\$
17	3, 16	mpbiran2 954	. . . . . . 7
18		opelxp 4799	. . . . . . . 8 $/\$
19	9	bianfi 960	. . . . . . . 8 $/\$
20	18, 19	bitr4i 187	. . . . . . 7
21	17, 20	orbi12i 776	. . . . . 6 $\/$ $\/$
22		elun 3370	. . . . . 6 $\/$
23	9	biorfi 758	. . . . . 6 $\/$
24	21, 22, 23	3bitr4ri 213	. . . . 5
25	1, 15, 24	3bitr4g 223	. . . 4
26	25	eqrdv 2236	. . 3
27		eleq2 2302	. . . . 5
28		opelxp 4799	. . . . . . . 8 $/\$
29		p0ex 4320	. . . . . . . . . . . 12
30	29	elsn 3721	. . . . . . . . . . 11
31		eqcom 2240	. . . . . . . . . . 11
32	30, 31	bitri 184	. . . . . . . . . 10
33	7, 32	nemtbir 2509	. . . . . . . . 9
34	33	bianfi 960	. . . . . . . 8 $/\$
35	28, 34	bitr4i 187	. . . . . . 7
36	29	snid 3736	. . . . . . . 8
37		opelxp 4799	. . . . . . . 8 $/\$
38	36, 37	mpbiran2 954	. . . . . . 7
39	35, 38	orbi12i 776	. . . . . 6 $\/$ $\/$
40		elun 3370	. . . . . 6 $\/$
41		biorf 756	. . . . . . 7 $\/$
42	33, 41	ax-mp 5	. . . . . 6 $\/$
43	39, 40, 42	3bitr4ri 213	. . . . 5
44		opelxp 4799	. . . . . . . 8 $/\$
45	33	bianfi 960	. . . . . . . 8 $/\$
46	44, 45	bitr4i 187	. . . . . . 7
47		opelxp 4799	. . . . . . . 8 $/\$
48	36, 47	mpbiran2 954	. . . . . . 7
49	46, 48	orbi12i 776	. . . . . 6 $\/$ $\/$
50		elun 3370	. . . . . 6 $\/$
51		biorf 756	. . . . . . 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 2236	. . 3
56	26, 55	jca 306	. 2 $/\$
57		xpeq1 4783	. . 3
58		xpeq1 4783	. . 3
59		uneq12 3378	. . 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 720

wceq 1402

wcel 2209

cun 3218

c0 3520

csn 3705

cop 3708

cxp 4767

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-opab 4188 df-xp 4775

This theorem is referenced by: (None)

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