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Mirrors > Home > NFE 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 2862 | . . . . 5 | |
2 | 1 | elintrab 3938 | . . . 4 |
3 | ssid 3290 | . . . . 5 | |
4 | sseq2 3293 | . . . . . . 7 | |
5 | eleq2 2414 | . . . . . . 7 | |
6 | 4, 5 | imbi12d 311 | . . . . . 6 |
7 | 6 | rspcv 2951 | . . . . 5 |
8 | 3, 7 | mpii 39 | . . . 4 |
9 | 2, 8 | syl5bi 208 | . . 3 |
10 | 9 | ssrdv 3278 | . 2 |
11 | ssintub 3944 | . . 3 | |
12 | 11 | a1i 10 | . 2 |
13 | 10, 12 | eqssd 3289 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1642 wcel 1710 wral 2614 crab 2618 wss 3257 cint 3926 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-int 3927 |
This theorem is referenced by: intmin2 3953 |
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