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Mirrors > Home > MPE Home > Th. List > 0elunit | Structured version Visualization version GIF version |
Description: Zero is an element of the closed unit interval. (Contributed by Scott Fenton, 11-Jun-2013.) |
Ref | Expression |
---|---|
0elunit | ⊢ 0 ∈ (0[,]1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10631 | . 2 ⊢ 0 ∈ ℝ | |
2 | 0le0 11726 | . 2 ⊢ 0 ≤ 0 | |
3 | 0le1 11151 | . 2 ⊢ 0 ≤ 1 | |
4 | elicc01 12842 | . 2 ⊢ (0 ∈ (0[,]1) ↔ (0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 ≤ 1)) | |
5 | 1, 2, 3, 4 | mpbir3an 1333 | 1 ⊢ 0 ∈ (0[,]1) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 ≤ cle 10664 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-icc 12733 |
This theorem is referenced by: xrhmeo 23477 htpycom 23507 htpyid 23508 htpyco1 23509 htpyco2 23510 htpycc 23511 phtpy01 23516 phtpycom 23519 phtpyid 23520 phtpyco2 23521 phtpycc 23522 reparphti 23528 pcocn 23548 pcohtpylem 23550 pcoptcl 23552 pcopt 23553 pcopt2 23554 pcoass 23555 pcorevcl 23556 pcorevlem 23557 pi1xfrf 23584 pi1xfr 23586 pi1xfrcnvlem 23587 pi1xfrcnv 23588 pi1cof 23590 pi1coghm 23592 dvlipcn 24518 lgamgulmlem2 25534 ttgcontlem1 26598 brbtwn2 26618 axsegconlem1 26630 axpaschlem 26653 axcontlem7 26683 axcontlem8 26684 xrge0iifcnv 31075 xrge0iifiso 31077 xrge0iifhom 31079 cnpconn 32374 pconnconn 32375 txpconn 32376 ptpconn 32377 indispconn 32378 connpconn 32379 sconnpi1 32383 txsconnlem 32384 txsconn 32385 cvxpconn 32386 cvxsconn 32387 cvmliftlem14 32441 cvmlift2lem2 32448 cvmlift2lem3 32449 cvmlift2lem8 32454 cvmlift2lem12 32458 cvmlift2lem13 32459 cvmliftphtlem 32461 cvmliftpht 32462 cvmlift3lem1 32463 cvmlift3lem2 32464 cvmlift3lem4 32466 cvmlift3lem5 32467 cvmlift3lem6 32468 cvmlift3lem9 32471 |
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