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Mirrors > Home > MPE Home > Th. List > ltneii | Structured version Visualization version GIF version |
Description: 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
ltneii.2 | ⊢ 𝐴 < 𝐵 |
Ref | Expression |
---|---|
ltneii | ⊢ 𝐴 ≠ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
2 | ltneii.2 | . . 3 ⊢ 𝐴 < 𝐵 | |
3 | 1, 2 | gtneii 10475 | . 2 ⊢ 𝐵 ≠ 𝐴 |
4 | 3 | necomi 3053 | 1 ⊢ 𝐴 ≠ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2164 ≠ wne 2999 class class class wbr 4875 ℝcr 10258 < clt 10398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-pre-lttri 10333 ax-pre-lttrn 10334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-ltxr 10403 |
This theorem is referenced by: 1ne2 11573 f1oun2prg 14045 geo2sum 14985 3dvds 15436 plusgndxnmulrndx 16364 basendxnmulrndx 16365 slotsbhcdif 16440 oppchomfval 16733 oppcbas 16737 rescbas 16848 rescabs 16852 odubas 17493 opprlem 18989 rmodislmod 19294 srasca 19549 sravsca 19550 opsrbaslem 19845 cnfldfun 20125 zlmlem 20232 zlmsca 20236 znbaslem 20253 thlbas 20410 thlle 20411 matbas 20593 matplusg 20594 tuslem 22448 setsmsbas 22657 tnglem 22821 ppiub 25349 2lgslem3 25549 2lgslem4 25551 ttgval 26181 ttglem 26182 slotsbaseefdif 26300 structvtxvallem 26325 usgrexmpldifpr 26562 upgr4cycl4dv4e 27557 konigsbergiedgw 27623 konigsberglem3 27629 konigsberglem5 27631 ex-dif 27834 ex-id 27845 ex-fv 27854 ex-mod 27860 resvbas 30373 resvplusg 30374 resvmulr 30376 hlhilslem 38008 rabren3dioph 38218 xrlexaddrp 40359 fourierdlem102 41213 fourierdlem114 41225 fouriersw 41236 nnsum4primesodd 42528 nnsum4primesoddALTV 42529 zlmodzxznm 43147 2p2ne5 43450 |
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