Theorem gtned 8255

Description: 'Less than' implies not equal. See also gtapd 8780 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

Step	Hyp	Ref	Expression
1		ltd.1	. 2 |- ( ph -> A e. RR )
2		ltned.2	. 2 |- ( ph -> A < B )
3		ltne 8227	. 2 |- ( ( A e. RR /\ A < B ) -> B =/= A )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> B =/= A )

Syntax hints:   -> wi 4   e. wcel 2200   =/= wne 2400   class class class wbr 4082   RRcr 7994   < clt 8177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-pre-ltirr 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4724  df-pnf 8179  df-mnf 8180  df-ltxr 8182

This theorem is referenced by:  ltned  8256  seq3f1olemqsumkj  10728  seqf1oglem1  10736  seqf1oglem2  10737  nn0opthlem2d  10938  zfz1isolemiso  11056  ennnfonelemim  12990  logbgcd1irr  15635  logbgcd1irraplemexp  15636  perfectlem2  15668  gausslemma2dlem4  15737