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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 2740 |
. . . . 5
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2 | 1 | elintrab 3856 |
. . . 4
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3 | ssid 3175 |
. . . . 5
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4 | sseq2 3179 |
. . . . . . 7
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5 | eleq2 2241 |
. . . . . . 7
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6 | 4, 5 | imbi12d 234 |
. . . . . 6
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7 | 6 | rspcv 2837 |
. . . . 5
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8 | 3, 7 | mpii 44 |
. . . 4
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9 | 2, 8 | biimtrid 152 |
. . 3
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10 | 9 | ssrdv 3161 |
. 2
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11 | ssintub 3862 |
. . 3
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12 | 11 | a1i 9 |
. 2
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13 | 10, 12 | eqssd 3172 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rab 2464 df-v 2739 df-in 3135 df-ss 3142 df-int 3845 |
This theorem is referenced by: intmin2 3870 bm2.5ii 4495 onsucmin 4506 cldcls 13584 |
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