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Mirrors > Home > ILE Home > Th. List > dfinfre | Unicode version |
Description: The infimum of a set of reals . (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
Ref | Expression |
---|---|
dfinfre | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6958 | . 2 inf | |
2 | df-sup 6957 | . . 3 | |
3 | ssel2 3142 | . . . . . . . . . 10 | |
4 | vex 2733 | . . . . . . . . . . . . 13 | |
5 | vex 2733 | . . . . . . . . . . . . 13 | |
6 | 4, 5 | brcnv 4792 | . . . . . . . . . . . 12 |
7 | 6 | notbii 663 | . . . . . . . . . . 11 |
8 | lenlt 7982 | . . . . . . . . . . 11 | |
9 | 7, 8 | bitr4id 198 | . . . . . . . . . 10 |
10 | 3, 9 | sylan2 284 | . . . . . . . . 9 |
11 | 10 | ancoms 266 | . . . . . . . 8 |
12 | 11 | an32s 563 | . . . . . . 7 |
13 | 12 | ralbidva 2466 | . . . . . 6 |
14 | 5, 4 | brcnv 4792 | . . . . . . . . 9 |
15 | vex 2733 | . . . . . . . . . . 11 | |
16 | 5, 15 | brcnv 4792 | . . . . . . . . . 10 |
17 | 16 | rexbii 2477 | . . . . . . . . 9 |
18 | 14, 17 | imbi12i 238 | . . . . . . . 8 |
19 | 18 | ralbii 2476 | . . . . . . 7 |
20 | 19 | a1i 9 | . . . . . 6 |
21 | 13, 20 | anbi12d 470 | . . . . 5 |
22 | 21 | rabbidva 2718 | . . . 4 |
23 | 22 | unieqd 3805 | . . 3 |
24 | 2, 23 | eqtrid 2215 | . 2 |
25 | 1, 24 | eqtrid 2215 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 wrex 2449 crab 2452 wss 3121 cuni 3794 class class class wbr 3987 ccnv 4608 csup 6955 infcinf 6956 cr 7760 clt 7941 cle 7942 |
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-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-cnv 4617 df-sup 6957 df-inf 6958 df-xr 7945 df-le 7947 |
This theorem is referenced by: (None) |
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