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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 2560 | . . . . 5 | |
2 | 1 | elintrab 3627 | . . . 4 |
3 | ssid 2964 | . . . . 5 | |
4 | sseq2 2967 | . . . . . . 7 | |
5 | eleq2 2101 | . . . . . . 7 | |
6 | 4, 5 | imbi12d 223 | . . . . . 6 |
7 | 6 | rspcv 2652 | . . . . 5 |
8 | 3, 7 | mpii 39 | . . . 4 |
9 | 2, 8 | syl5bi 141 | . . 3 |
10 | 9 | ssrdv 2951 | . 2 |
11 | ssintub 3633 | . . 3 | |
12 | 11 | a1i 9 | . 2 |
13 | 10, 12 | eqssd 2962 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1243 wcel 1393 wral 2306 crab 2310 wss 2917 cint 3615 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rab 2315 df-v 2559 df-in 2924 df-ss 2931 df-int 3616 |
This theorem is referenced by: intmin2 3641 bm2.5ii 4222 onsucmin 4233 |
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