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Mirrors > Home > MPE Home > Th. List > opthreg | Structured version Unicode version |
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 7919 (via the preleq 7935 step). See df-op 3993 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
Ref | Expression |
---|---|
preleq.1 |
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preleq.2 |
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preleq.3 |
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preleq.4 |
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Ref | Expression |
---|---|
opthreg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preleq.1 |
. . . . 5
![]() ![]() ![]() ![]() | |
2 | 1 | prid1 4092 |
. . . 4
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3 | preleq.3 |
. . . . 5
![]() ![]() ![]() ![]() | |
4 | 3 | prid1 4092 |
. . . 4
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5 | prex 4643 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | prex 4643 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 1, 5, 3, 6 | preleq 7935 |
. . . 4
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8 | 2, 4, 7 | mpanl12 682 |
. . 3
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9 | preq1 4063 |
. . . . . 6
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10 | 9 | eqeq1d 2456 |
. . . . 5
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11 | preleq.2 |
. . . . . 6
![]() ![]() ![]() ![]() | |
12 | preleq.4 |
. . . . . 6
![]() ![]() ![]() ![]() | |
13 | 11, 12 | preqr2 4156 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 10, 13 | syl6bi 228 |
. . . 4
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15 | 14 | imdistani 690 |
. . 3
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16 | 8, 15 | syl 16 |
. 2
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17 | preq1 4063 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 17 | adantr 465 |
. . 3
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19 | preq12 4065 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | preq2d 4070 |
. . 3
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21 | 18, 20 | eqtrd 2495 |
. 2
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22 | 16, 21 | impbii 188 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4522 ax-nul 4530 ax-pr 4640 ax-reg 7919 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-nul 3747 df-if 3901 df-sn 3987 df-pr 3989 df-op 3993 df-br 4402 df-opab 4460 df-eprel 4741 df-fr 4788 |
This theorem is referenced by: (None) |
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