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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 4452 (via the preleq 4470 step). See df-op 3536 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 | |
preleq.3 | |
preleq.4 |
Ref | Expression |
---|---|
opthreg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preleq.1 | . . . . 5 | |
2 | 1 | prid1 3629 | . . . 4 |
3 | preleq.3 | . . . . 5 | |
4 | 3 | prid1 3629 | . . . 4 |
5 | preleq.2 | . . . . . 6 | |
6 | prexg 4133 | . . . . . 6 | |
7 | 1, 5, 6 | mp2an 422 | . . . . 5 |
8 | preleq.4 | . . . . . 6 | |
9 | prexg 4133 | . . . . . 6 | |
10 | 3, 8, 9 | mp2an 422 | . . . . 5 |
11 | 1, 7, 3, 10 | preleq 4470 | . . . 4 |
12 | 2, 4, 11 | mpanl12 432 | . . 3 |
13 | preq1 3600 | . . . . . 6 | |
14 | 13 | eqeq1d 2148 | . . . . 5 |
15 | 5, 8 | preqr2 3696 | . . . . 5 |
16 | 14, 15 | syl6bi 162 | . . . 4 |
17 | 16 | imdistani 441 | . . 3 |
18 | 12, 17 | syl 14 | . 2 |
19 | preq1 3600 | . . . 4 | |
20 | 19 | adantr 274 | . . 3 |
21 | preq12 3602 | . . . 4 | |
22 | 21 | preq2d 3607 | . . 3 |
23 | 20, 22 | eqtrd 2172 | . 2 |
24 | 18, 23 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wcel 1480 cvv 2686 cpr 3528 |
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-pr 4131 ax-setind 4452 |
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-ral 2421 df-v 2688 df-dif 3073 df-un 3075 df-sn 3533 df-pr 3534 |
This theorem is referenced by: (None) |
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