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Mirrors > Home > ILE Home > Th. List > opthprc | Unicode version |
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.) |
Ref | Expression |
---|---|
opthprc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2203 | . . . . 5 | |
2 | 0ex 4055 | . . . . . . . . 9 | |
3 | 2 | snid 3556 | . . . . . . . 8 |
4 | opelxp 4569 | . . . . . . . 8 | |
5 | 3, 4 | mpbiran2 925 | . . . . . . 7 |
6 | opelxp 4569 | . . . . . . . 8 | |
7 | 0nep0 4089 | . . . . . . . . . 10 | |
8 | 2 | elsn 3543 | . . . . . . . . . 10 |
9 | 7, 8 | nemtbir 2397 | . . . . . . . . 9 |
10 | 9 | bianfi 931 | . . . . . . . 8 |
11 | 6, 10 | bitr4i 186 | . . . . . . 7 |
12 | 5, 11 | orbi12i 753 | . . . . . 6 |
13 | elun 3217 | . . . . . 6 | |
14 | 9 | biorfi 735 | . . . . . 6 |
15 | 12, 13, 14 | 3bitr4ri 212 | . . . . 5 |
16 | opelxp 4569 | . . . . . . . 8 | |
17 | 3, 16 | mpbiran2 925 | . . . . . . 7 |
18 | opelxp 4569 | . . . . . . . 8 | |
19 | 9 | bianfi 931 | . . . . . . . 8 |
20 | 18, 19 | bitr4i 186 | . . . . . . 7 |
21 | 17, 20 | orbi12i 753 | . . . . . 6 |
22 | elun 3217 | . . . . . 6 | |
23 | 9 | biorfi 735 | . . . . . 6 |
24 | 21, 22, 23 | 3bitr4ri 212 | . . . . 5 |
25 | 1, 15, 24 | 3bitr4g 222 | . . . 4 |
26 | 25 | eqrdv 2137 | . . 3 |
27 | eleq2 2203 | . . . . 5 | |
28 | opelxp 4569 | . . . . . . . 8 | |
29 | p0ex 4112 | . . . . . . . . . . . 12 | |
30 | 29 | elsn 3543 | . . . . . . . . . . 11 |
31 | eqcom 2141 | . . . . . . . . . . 11 | |
32 | 30, 31 | bitri 183 | . . . . . . . . . 10 |
33 | 7, 32 | nemtbir 2397 | . . . . . . . . 9 |
34 | 33 | bianfi 931 | . . . . . . . 8 |
35 | 28, 34 | bitr4i 186 | . . . . . . 7 |
36 | 29 | snid 3556 | . . . . . . . 8 |
37 | opelxp 4569 | . . . . . . . 8 | |
38 | 36, 37 | mpbiran2 925 | . . . . . . 7 |
39 | 35, 38 | orbi12i 753 | . . . . . 6 |
40 | elun 3217 | . . . . . 6 | |
41 | biorf 733 | . . . . . . 7 | |
42 | 33, 41 | ax-mp 5 | . . . . . 6 |
43 | 39, 40, 42 | 3bitr4ri 212 | . . . . 5 |
44 | opelxp 4569 | . . . . . . . 8 | |
45 | 33 | bianfi 931 | . . . . . . . 8 |
46 | 44, 45 | bitr4i 186 | . . . . . . 7 |
47 | opelxp 4569 | . . . . . . . 8 | |
48 | 36, 47 | mpbiran2 925 | . . . . . . 7 |
49 | 46, 48 | orbi12i 753 | . . . . . 6 |
50 | elun 3217 | . . . . . 6 | |
51 | biorf 733 | . . . . . . 7 | |
52 | 33, 51 | ax-mp 5 | . . . . . 6 |
53 | 49, 50, 52 | 3bitr4ri 212 | . . . . 5 |
54 | 27, 43, 53 | 3bitr4g 222 | . . . 4 |
55 | 54 | eqrdv 2137 | . . 3 |
56 | 26, 55 | jca 304 | . 2 |
57 | xpeq1 4553 | . . 3 | |
58 | xpeq1 4553 | . . 3 | |
59 | uneq12 3225 | . . 3 | |
60 | 57, 58, 59 | syl2an 287 | . 2 |
61 | 56, 60 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 cun 3069 c0 3363 csn 3527 cop 3530 cxp 4537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-xp 4545 |
This theorem is referenced by: (None) |
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