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Mirrors > Home > ILE Home > Th. List > qfto | Unicode version |
Description: A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.) |
Ref | Expression |
---|---|
qfto |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 4467 |
. . . 4
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2 | brun 3891 |
. . . . 5
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3 | df-br 3846 |
. . . . 5
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4 | vex 2622 |
. . . . . . 7
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5 | vex 2622 |
. . . . . . 7
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6 | 4, 5 | brcnv 4619 |
. . . . . 6
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7 | 6 | orbi2i 714 |
. . . . 5
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8 | 2, 3, 7 | 3bitr3i 208 |
. . . 4
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9 | 1, 8 | imbi12i 237 |
. . 3
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10 | 9 | 2albii 1405 |
. 2
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11 | relxp 4547 |
. . 3
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12 | ssrel 4526 |
. . 3
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13 | 11, 12 | ax-mp 7 |
. 2
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14 | r2al 2397 |
. 2
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15 | 10, 13, 14 | 3bitr4i 210 |
1
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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 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-xp 4444 df-rel 4445 df-cnv 4446 |
This theorem is referenced by: (None) |
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