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Theorem ltneOLD 9678
Description: 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ltneOLD  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  =/=  A )

Proof of Theorem ltneOLD
StepHypRef Expression
1 ltne 9677 . 2  |-  ( ( A  e.  RR  /\  A  <  B )  ->  B  =/=  A )
213adant2 1015 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  =/=  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    e. wcel 1767    =/= wne 2662   class class class wbr 4447   RRcr 9487    < clt 9624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-pre-lttri 9562  ax-pre-lttrn 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-ltxr 9629
This theorem is referenced by:  ltlen  9682  znnenlem  13802  coprm  14096  phibndlem  14155  sineq0  22647  stadd3i  26843
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