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Theorem gtned 8021
Description: 'Less than' implies not equal. See also gtapd 8545 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
ltd.1  |-  ( ph  ->  A  e.  RR )
ltned.2  |-  ( ph  ->  A  <  B )
Assertion
Ref Expression
gtned  |-  ( ph  ->  B  =/=  A )

Proof of Theorem gtned
StepHypRef Expression
1 ltd.1 . 2  |-  ( ph  ->  A  e.  RR )
2 ltned.2 . 2  |-  ( ph  ->  A  <  B )
3 ltne 7993 . 2  |-  ( ( A  e.  RR  /\  A  <  B )  ->  B  =/=  A )
41, 2, 3syl2anc 409 1  |-  ( ph  ->  B  =/=  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2141    =/= wne 2340   class class class wbr 3987   RRcr 7762    < clt 7943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7854  ax-resscn 7855  ax-pre-ltirr 7875
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-xp 4615  df-pnf 7945  df-mnf 7946  df-ltxr 7948
This theorem is referenced by:  ltned  8022  seq3f1olemqsumkj  10443  nn0opthlem2d  10644  zfz1isolemiso  10763  ennnfonelemim  12368  logbgcd1irr  13640  logbgcd1irraplemexp  13641
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