opthprc - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > opthprc		Unicode version

Theorem opthprc 4391

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 2101	. . . . 5
2		0ex 3884	. . . . . . . . 9
3	2	snid 3402	. . . . . . . 8
4		opelxp 4374	. . . . . . . 8 $/\$
5	3, 4	mpbiran2 848	. . . . . . 7
6		opelxp 4374	. . . . . . . 8 $/\$
7		0nep0 3918	. . . . . . . . . 10
8	2	elsn 3391	. . . . . . . . . 10
9	7, 8	nemtbir 2294	. . . . . . . . 9
10	9	bianfi 854	. . . . . . . 8 $/\$
11	6, 10	bitr4i 176	. . . . . . 7
12	5, 11	orbi12i 681	. . . . . 6 $\/$ $\/$
13		elun 3084	. . . . . 6 $\/$
14	9	biorfi 665	. . . . . 6 $\/$
15	12, 13, 14	3bitr4ri 202	. . . . 5
16		opelxp 4374	. . . . . . . 8 $/\$
17	3, 16	mpbiran2 848	. . . . . . 7
18		opelxp 4374	. . . . . . . 8 $/\$
19	9	bianfi 854	. . . . . . . 8 $/\$
20	18, 19	bitr4i 176	. . . . . . 7
21	17, 20	orbi12i 681	. . . . . 6 $\/$ $\/$
22		elun 3084	. . . . . 6 $\/$
23	9	biorfi 665	. . . . . 6 $\/$
24	21, 22, 23	3bitr4ri 202	. . . . 5
25	1, 15, 24	3bitr4g 212	. . . 4
26	25	eqrdv 2038	. . 3
27		eleq2 2101	. . . . 5
28		opelxp 4374	. . . . . . . 8 $/\$
29		p0ex 3939	. . . . . . . . . . . 12
30	29	elsn 3391	. . . . . . . . . . 11
31		eqcom 2042	. . . . . . . . . . 11
32	30, 31	bitri 173	. . . . . . . . . 10
33	7, 32	nemtbir 2294	. . . . . . . . 9
34	33	bianfi 854	. . . . . . . 8 $/\$
35	28, 34	bitr4i 176	. . . . . . 7
36	29	snid 3402	. . . . . . . 8
37		opelxp 4374	. . . . . . . 8 $/\$
38	36, 37	mpbiran2 848	. . . . . . 7
39	35, 38	orbi12i 681	. . . . . 6 $\/$ $\/$
40		elun 3084	. . . . . 6 $\/$
41		biorf 663	. . . . . . 7 $\/$
42	33, 41	ax-mp 7	. . . . . 6 $\/$
43	39, 40, 42	3bitr4ri 202	. . . . 5
44		opelxp 4374	. . . . . . . 8 $/\$
45	33	bianfi 854	. . . . . . . 8 $/\$
46	44, 45	bitr4i 176	. . . . . . 7
47		opelxp 4374	. . . . . . . 8 $/\$
48	36, 47	mpbiran2 848	. . . . . . 7
49	46, 48	orbi12i 681	. . . . . 6 $\/$ $\/$
50		elun 3084	. . . . . 6 $\/$
51		biorf 663	. . . . . . 7 $\/$
52	33, 51	ax-mp 7	. . . . . 6 $\/$
53	49, 50, 52	3bitr4ri 202	. . . . 5
54	27, 43, 53	3bitr4g 212	. . . 4
55	54	eqrdv 2038	. . 3
56	26, 55	jca 290	. 2 $/\$
57		xpeq1 4359	. . 3
58		xpeq1 4359	. . 3
59		uneq12 3092	. . 3 $/\$
60	57, 58, 59	syl2an 273	. 2 $/\$
61	56, 60	impbii 117	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 97

wb 98 $\/$ wo 629

wceq 1243

wcel 1393

cun 2915

c0 3224

csn 3375

cop 3378

cxp 4343

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351

This theorem is referenced by: (None)

W3C validator