Theorem gtned 8205

Description: 'Less than' implies not equal. See also gtapd 8730 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 8177	. 2 |- ( ( A e. RR /\ A < B ) -> B =/= A )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> B =/= A )

Syntax hints:   -> wi 4   e. wcel 2177   =/= wne 2377   class class class wbr 4051   RRcr 7944   < clt 8127

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-pre-ltirr 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-xp 4689  df-pnf 8129  df-mnf 8130  df-ltxr 8132

This theorem is referenced by:  ltned  8206  seq3f1olemqsumkj  10678  seqf1oglem1  10686  seqf1oglem2  10687  nn0opthlem2d  10888  zfz1isolemiso  11006  ennnfonelemim  12870  logbgcd1irr  15514  logbgcd1irraplemexp  15515  perfectlem2  15547  gausslemma2dlem4  15616