Theorem gtned 8167

Description: 'Less than' implies not equal. See also gtapd 8692 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 8139	. 2 |- ( ( A e. RR /\ A < B ) -> B =/= A )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> B =/= A )

Syntax hints:   -> wi 4   e. wcel 2175   =/= wne 2375   class class class wbr 4043   RRcr 7906   < clt 8089

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-pre-ltirr 8019

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-xp 4679  df-pnf 8091  df-mnf 8092  df-ltxr 8094

This theorem is referenced by:  ltned  8168  seq3f1olemqsumkj  10637  seqf1oglem1  10645  seqf1oglem2  10646  nn0opthlem2d  10847  zfz1isolemiso  10965  ennnfonelemim  12714  logbgcd1irr  15357  logbgcd1irraplemexp  15358  perfectlem2  15390  gausslemma2dlem4  15459