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Mirrors > Home > ILE Home > Th. List > intmin | Unicode version |
Description: Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
intmin |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2622 |
. . . . 5
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2 | 1 | elintrab 3700 |
. . . 4
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3 | ssid 3044 |
. . . . 5
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4 | sseq2 3048 |
. . . . . . 7
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5 | eleq2 2151 |
. . . . . . 7
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6 | 4, 5 | imbi12d 232 |
. . . . . 6
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7 | 6 | rspcv 2718 |
. . . . 5
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8 | 3, 7 | mpii 43 |
. . . 4
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9 | 2, 8 | syl5bi 150 |
. . 3
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10 | 9 | ssrdv 3031 |
. 2
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11 | ssintub 3706 |
. . 3
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12 | 11 | a1i 9 |
. 2
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13 | 10, 12 | eqssd 3042 |
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-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rab 2368 df-v 2621 df-in 3005 df-ss 3012 df-int 3689 |
This theorem is referenced by: intmin2 3714 bm2.5ii 4313 onsucmin 4324 |
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