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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 2715 | . . . . 5 | |
2 | 1 | elintrab 3820 | . . . 4 |
3 | ssid 3148 | . . . . 5 | |
4 | sseq2 3152 | . . . . . . 7 | |
5 | eleq2 2221 | . . . . . . 7 | |
6 | 4, 5 | imbi12d 233 | . . . . . 6 |
7 | 6 | rspcv 2812 | . . . . 5 |
8 | 3, 7 | mpii 44 | . . . 4 |
9 | 2, 8 | syl5bi 151 | . . 3 |
10 | 9 | ssrdv 3134 | . 2 |
11 | ssintub 3826 | . . 3 | |
12 | 11 | a1i 9 | . 2 |
13 | 10, 12 | eqssd 3145 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1335 wcel 2128 wral 2435 crab 2439 wss 3102 cint 3808 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rab 2444 df-v 2714 df-in 3108 df-ss 3115 df-int 3809 |
This theorem is referenced by: intmin2 3834 bm2.5ii 4456 onsucmin 4467 cldcls 12556 |
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