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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 4519 (via the preleq 4537 step). See df-op 3590 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 3687 | . . . 4 |
3 | preleq.3 | . . . . 5 | |
4 | 3 | prid1 3687 | . . . 4 |
5 | preleq.2 | . . . . . 6 | |
6 | prexg 4194 | . . . . . 6 | |
7 | 1, 5, 6 | mp2an 424 | . . . . 5 |
8 | preleq.4 | . . . . . 6 | |
9 | prexg 4194 | . . . . . 6 | |
10 | 3, 8, 9 | mp2an 424 | . . . . 5 |
11 | 1, 7, 3, 10 | preleq 4537 | . . . 4 |
12 | 2, 4, 11 | mpanl12 434 | . . 3 |
13 | preq1 3658 | . . . . . 6 | |
14 | 13 | eqeq1d 2179 | . . . . 5 |
15 | 5, 8 | preqr2 3754 | . . . . 5 |
16 | 14, 15 | syl6bi 162 | . . . 4 |
17 | 16 | imdistani 443 | . . 3 |
18 | 12, 17 | syl 14 | . 2 |
19 | preq1 3658 | . . . 4 | |
20 | 19 | adantr 274 | . . 3 |
21 | preq12 3660 | . . . 4 | |
22 | 21 | preq2d 3665 | . . 3 |
23 | 20, 22 | eqtrd 2203 | . 2 |
24 | 18, 23 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1348 wcel 2141 cvv 2730 cpr 3582 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pr 4192 ax-setind 4519 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-dif 3123 df-un 3125 df-sn 3587 df-pr 3588 |
This theorem is referenced by: (None) |
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