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Mirrors > Home > MPE Home > Th. List > ltnei | Structured version Unicode version |
Description: 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.) |
Ref | Expression |
---|---|
lt.1 |
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lt.2 |
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Ref | Expression |
---|---|
ltnei |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.1 |
. 2
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2 | lt.2 |
. 2
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3 | ltneOLD 9575 |
. 2
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4 | 1, 2, 3 | mp3an12 1305 |
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 1952 ax-ext 2430 ax-sep 4513 ax-nul 4521 ax-pow 4570 ax-pr 4631 ax-un 6474 ax-resscn 9442 ax-pre-lttri 9459 ax-pre-lttrn 9460 |
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 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-rab 2804 df-v 3072 df-sbc 3287 df-csb 3389 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-nul 3738 df-if 3892 df-pw 3962 df-sn 3978 df-pr 3980 df-op 3984 df-uni 4192 df-br 4393 df-opab 4451 df-mpt 4452 df-id 4736 df-po 4741 df-so 4742 df-xp 4946 df-rel 4947 df-cnv 4948 df-co 4949 df-dm 4950 df-rn 4951 df-res 4952 df-ima 4953 df-iota 5481 df-fun 5520 df-fn 5521 df-f 5522 df-f1 5523 df-fo 5524 df-f1o 5525 df-fv 5526 df-er 7203 df-en 7413 df-dom 7414 df-sdom 7415 df-pnf 9523 df-mnf 9524 df-ltxr 9526 |
This theorem is referenced by: gt0ne0i 9978 nn0opthlem2 12150 hashtpg 12290 sralem 17366 cchhllem 23270 sgnnbi 27064 sgnpbi 27065 signswch 27098 |
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