opthprc - Intuitionistic Logic Explorer

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

Theorem opthprc 4744

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 2271	. . . . 5
2		0ex 4187	. . . . . . . . 9
3	2	snid 3674	. . . . . . . 8
4		opelxp 4723	. . . . . . . 8 $/\$
5	3, 4	mpbiran2 944	. . . . . . 7
6		opelxp 4723	. . . . . . . 8 $/\$
7		0nep0 4225	. . . . . . . . . 10
8	2	elsn 3659	. . . . . . . . . 10
9	7, 8	nemtbir 2467	. . . . . . . . 9
10	9	bianfi 950	. . . . . . . 8 $/\$
11	6, 10	bitr4i 187	. . . . . . 7
12	5, 11	orbi12i 766	. . . . . 6 $\/$ $\/$
13		elun 3322	. . . . . 6 $\/$
14	9	biorfi 748	. . . . . 6 $\/$
15	12, 13, 14	3bitr4ri 213	. . . . 5
16		opelxp 4723	. . . . . . . 8 $/\$
17	3, 16	mpbiran2 944	. . . . . . 7
18		opelxp 4723	. . . . . . . 8 $/\$
19	9	bianfi 950	. . . . . . . 8 $/\$
20	18, 19	bitr4i 187	. . . . . . 7
21	17, 20	orbi12i 766	. . . . . 6 $\/$ $\/$
22		elun 3322	. . . . . 6 $\/$
23	9	biorfi 748	. . . . . 6 $\/$
24	21, 22, 23	3bitr4ri 213	. . . . 5
25	1, 15, 24	3bitr4g 223	. . . 4
26	25	eqrdv 2205	. . 3
27		eleq2 2271	. . . . 5
28		opelxp 4723	. . . . . . . 8 $/\$
29		p0ex 4248	. . . . . . . . . . . 12
30	29	elsn 3659	. . . . . . . . . . 11
31		eqcom 2209	. . . . . . . . . . 11
32	30, 31	bitri 184	. . . . . . . . . 10
33	7, 32	nemtbir 2467	. . . . . . . . 9
34	33	bianfi 950	. . . . . . . 8 $/\$
35	28, 34	bitr4i 187	. . . . . . 7
36	29	snid 3674	. . . . . . . 8
37		opelxp 4723	. . . . . . . 8 $/\$
38	36, 37	mpbiran2 944	. . . . . . 7
39	35, 38	orbi12i 766	. . . . . 6 $\/$ $\/$
40		elun 3322	. . . . . 6 $\/$
41		biorf 746	. . . . . . 7 $\/$
42	33, 41	ax-mp 5	. . . . . 6 $\/$
43	39, 40, 42	3bitr4ri 213	. . . . 5
44		opelxp 4723	. . . . . . . 8 $/\$
45	33	bianfi 950	. . . . . . . 8 $/\$
46	44, 45	bitr4i 187	. . . . . . 7
47		opelxp 4723	. . . . . . . 8 $/\$
48	36, 47	mpbiran2 944	. . . . . . 7
49	46, 48	orbi12i 766	. . . . . 6 $\/$ $\/$
50		elun 3322	. . . . . 6 $\/$
51		biorf 746	. . . . . . 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 2205	. . 3
56	26, 55	jca 306	. 2 $/\$
57		xpeq1 4707	. . 3
58		xpeq1 4707	. . 3
59		uneq12 3330	. . 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 710

wceq 1373

wcel 2178

cun 3172

c0 3468

csn 3643

cop 3646

cxp 4691

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-ral 2491 df-rex 2492 df-v 2778 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-opab 4122 df-xp 4699

This theorem is referenced by: (None)

W3C validator