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Mirrors > Home > ILE Home > Th. List > opthreg | Unicode version |
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4390 (via the preleq 4408 step). See df-op 3483 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
Ref | Expression |
---|---|
preleq.1 |
![]() ![]() ![]() ![]() |
preleq.2 |
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preleq.3 |
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preleq.4 |
![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
opthreg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preleq.1 |
. . . . 5
![]() ![]() ![]() ![]() | |
2 | 1 | prid1 3576 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | preleq.3 |
. . . . 5
![]() ![]() ![]() ![]() | |
4 | 3 | prid1 3576 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | preleq.2 |
. . . . . 6
![]() ![]() ![]() ![]() | |
6 | prexg 4071 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 1, 5, 6 | mp2an 420 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | preleq.4 |
. . . . . 6
![]() ![]() ![]() ![]() | |
9 | prexg 4071 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 3, 8, 9 | mp2an 420 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 1, 7, 3, 10 | preleq 4408 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 2, 4, 11 | mpanl12 430 |
. . 3
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13 | preq1 3547 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | eqeq1d 2108 |
. . . . 5
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15 | 5, 8 | preqr2 3643 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 14, 15 | syl6bi 162 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | imdistani 437 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 12, 17 | syl 14 |
. 2
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19 | preq1 3547 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | adantr 272 |
. . 3
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21 | preq12 3549 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 21 | preq2d 3554 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 20, 22 | eqtrd 2132 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 18, 23 | impbii 125 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pr 4069 ax-setind 4390 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-v 2643 df-dif 3023 df-un 3025 df-sn 3480 df-pr 3481 |
This theorem is referenced by: (None) |
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