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Mirrors > Home > MPE Home > Th. List > infmrgelb | Unicode version |
Description: Any lower bound of a nonempty set of real numbers is less than or equal to its infimum. (Contributed by Jeff Hankins, 1-Sep-2013.) (Revised by Mario Carneiro, 6-Sep-2014.) |
Ref | Expression |
---|---|
infmrgelb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gtso 9687 | . . . . . . 7 | |
2 | 1 | a1i 11 | . . . . . 6 |
3 | infm3 10527 | . . . . . . 7 | |
4 | vex 3112 | . . . . . . . . . . . 12 | |
5 | vex 3112 | . . . . . . . . . . . 12 | |
6 | 4, 5 | brcnv 5190 | . . . . . . . . . . 11 |
7 | 6 | notbii 296 | . . . . . . . . . 10 |
8 | 7 | ralbii 2888 | . . . . . . . . 9 |
9 | 5, 4 | brcnv 5190 | . . . . . . . . . . 11 |
10 | vex 3112 | . . . . . . . . . . . . 13 | |
11 | 5, 10 | brcnv 5190 | . . . . . . . . . . . 12 |
12 | 11 | rexbii 2959 | . . . . . . . . . . 11 |
13 | 9, 12 | imbi12i 326 | . . . . . . . . . 10 |
14 | 13 | ralbii 2888 | . . . . . . . . 9 |
15 | 8, 14 | anbi12i 697 | . . . . . . . 8 |
16 | 15 | rexbii 2959 | . . . . . . 7 |
17 | 3, 16 | sylibr 212 | . . . . . 6 |
18 | simp1 996 | . . . . . 6 | |
19 | 2, 17, 18 | suplub2 7941 | . . . . 5 |
20 | 19 | notbid 294 | . . . 4 |
21 | ralnex 2903 | . . . 4 | |
22 | 20, 21 | syl6bbr 263 | . . 3 |
23 | simpr 461 | . . . 4 | |
24 | infmrcl 10547 | . . . . 5 | |
25 | 24 | adantr 465 | . . . 4 |
26 | lenlt 9684 | . . . . 5 | |
27 | brcnvg 5188 | . . . . . 6 | |
28 | 27 | notbid 294 | . . . . 5 |
29 | 26, 28 | bitr4d 256 | . . . 4 |
30 | 23, 25, 29 | syl2anc 661 | . . 3 |
31 | 23 | adantr 465 | . . . . 5 |
32 | simpl1 999 | . . . . . 6 | |
33 | 32 | sselda 3503 | . . . . 5 |
34 | lenlt 9684 | . . . . . 6 | |
35 | brcnvg 5188 | . . . . . . 7 | |
36 | 35 | notbid 294 | . . . . . 6 |
37 | 34, 36 | bitr4d 256 | . . . . 5 |
38 | 31, 33, 37 | syl2anc 661 | . . . 4 |
39 | 38 | ralbidva 2893 | . . 3 |
40 | 22, 30, 39 | 3bitr4d 285 | . 2 |
41 | breq2 4456 | . . 3 | |
42 | 41 | cbvralv 3084 | . 2 |
43 | 40, 42 | syl6bb 261 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -. wn 3 -> wi 4
<-> wb 184 /\ wa 369 /\ w3a 973
e. wcel 1818 =/= wne 2652 A. wral 2807
E. wrex 2808 C_ wss 3475 c0 3784 class class class wbr 4452
Or wor 4804 `' ccnv 5003 sup csup 7920
cr 9512 clt 9649 cle 9650 |
This theorem is referenced by: infmxrre 11556 minveclem2 21841 minveclem3b 21843 minveclem4 21847 minveclem6 21849 pilem2 22847 pilem3 22848 pntlem3 23794 minvecolem2 25791 minvecolem4 25796 minvecolem5 25797 minvecolem6 25798 infmrgelbi 30814 taupi 37698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 ax-pre-mulgt0 9590 ax-pre-sup 9591 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-op 4036 df-uni 4250 df-br 4453 df-opab 4511 df-mpt 4512 df-id 4800 df-po 4805 df-so 4806 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-isom 5602 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-sup 7921 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-le 9655 df-sub 9830 df-neg 9831 |
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