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Mirrors > Home > MPE Home > Th. List > genpnmax | Structured version Unicode version |
Description: An operation on positive reals has no largest member. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
genp.1 |
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genp.2 |
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genpnmax.2 |
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genpnmax.3 |
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Ref | Expression |
---|---|
genpnmax |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | genp.1 |
. . 3
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2 | genp.2 |
. . 3
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3 | 1, 2 | genpelv 9284 |
. 2
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4 | prnmax 9279 |
. . . . . . . 8
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5 | 4 | adantr 465 |
. . . . . . 7
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6 | 1, 2 | genpprecl 9285 |
. . . . . . . . . . . . . . 15
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7 | 6 | exp4b 607 |
. . . . . . . . . . . . . 14
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8 | 7 | com34 83 |
. . . . . . . . . . . . 13
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9 | 8 | imp32 433 |
. . . . . . . . . . . 12
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10 | elprnq 9275 |
. . . . . . . . . . . . . 14
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11 | vex 3081 |
. . . . . . . . . . . . . . . 16
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12 | vex 3081 |
. . . . . . . . . . . . . . . 16
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13 | genpnmax.2 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | vex 3081 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() | |
15 | genpnmax.3 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 11, 12, 13, 14, 15 | caovord2 6388 |
. . . . . . . . . . . . . . 15
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17 | 16 | biimpd 207 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 10, 17 | syl 16 |
. . . . . . . . . . . . 13
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19 | 18 | adantl 466 |
. . . . . . . . . . . 12
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20 | 9, 19 | anim12d 563 |
. . . . . . . . . . 11
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21 | breq2 4407 |
. . . . . . . . . . . 12
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22 | 21 | rspcev 3179 |
. . . . . . . . . . 11
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23 | 20, 22 | syl6 33 |
. . . . . . . . . 10
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24 | 23 | adantlr 714 |
. . . . . . . . 9
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25 | 24 | expd 436 |
. . . . . . . 8
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26 | 25 | rexlimdv 2946 |
. . . . . . 7
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27 | 5, 26 | mpd 15 |
. . . . . 6
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28 | 27 | an4s 822 |
. . . . 5
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29 | breq1 4406 |
. . . . . 6
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30 | 29 | rexbidv 2868 |
. . . . 5
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31 | 28, 30 | syl5ibr 221 |
. . . 4
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32 | 31 | expdcom 439 |
. . 3
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33 | 32 | rexlimdvv 2953 |
. 2
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34 | 3, 33 | sylbid 215 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-inf2 7962 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-br 4404 df-opab 4462 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-iota 5492 df-fun 5531 df-fv 5537 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-om 6590 df-ni 9156 df-nq 9196 df-np 9265 |
This theorem is referenced by: genpcl 9292 |
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