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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2663 | . . . . 5 | |
2 | 1 | elintrab 3753 | . . . 4 |
3 | ssid 3087 | . . . . 5 | |
4 | sseq2 3091 | . . . . . . 7 | |
5 | eleq2 2181 | . . . . . . 7 | |
6 | 4, 5 | imbi12d 233 | . . . . . 6 |
7 | 6 | rspcv 2759 | . . . . 5 |
8 | 3, 7 | mpii 44 | . . . 4 |
9 | 2, 8 | syl5bi 151 | . . 3 |
10 | 9 | ssrdv 3073 | . 2 |
11 | ssintub 3759 | . . 3 | |
12 | 11 | a1i 9 | . 2 |
13 | 10, 12 | eqssd 3084 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 wcel 1465 wral 2393 crab 2397 wss 3041 cint 3741 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rab 2402 df-v 2662 df-in 3047 df-ss 3054 df-int 3742 |
This theorem is referenced by: intmin2 3767 bm2.5ii 4382 onsucmin 4393 cldcls 12210 |
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