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Mirrors > Home > MPE Home > Th. List > min1 | Structured version Visualization version Unicode version |
Description: The minimum of two numbers is less than or equal to the first. (Contributed by NM, 3-Aug-2007.) |
Ref | Expression |
---|---|
min1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 9712 |
. 2
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2 | rexr 9712 |
. 2
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3 | xrmin1 11501 |
. 2
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4 | 1, 2, 3 | syl2an 484 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610 ax-cnex 9621 ax-resscn 9622 ax-pre-lttri 9639 ax-pre-lttrn 9640 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-nel 2636 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-po 4774 df-so 4775 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-er 7389 df-en 7596 df-dom 7597 df-sdom 7598 df-pnf 9703 df-mnf 9704 df-xr 9705 df-ltxr 9706 df-le 9707 |
This theorem is referenced by: reccn2 13709 ssblex 21492 nlmvscnlem1 21738 nrginvrcnlem 21742 icccmplem2 21890 xlebnum 22045 ipcnlem1 22265 ivthlem2 22452 ioombl1lem4 22563 mbfi1fseqlem5 22726 aalioulem5 23341 aalioulem6 23342 logcnlem3 23638 cxpcn3lem 23736 ftalem5 24050 ftalem5OLD 24052 chtdif 24134 ppidif 24139 chebbnd1lem1 24356 itg2addnc 32041 mullimc 37734 mullimcf 37741 limcleqr 37763 addlimc 37767 0ellimcdiv 37768 limclner 37770 stoweidlem5 37903 fourierdlem103 38111 fourierdlem104 38112 hsphoidmvle 38446 hoidmv1lelem1 38451 hoidmv1lelem2 38452 hoidmv1lelem3 38453 |
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