Theorem ltne 8400

Description: 'Less than' implies not equal. See also ltap 8951 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)

Assertion

Ref	Expression
ltne	|- ( ( A e. RR /\ A < B ) -> B =/= A )

Proof of Theorem ltne

Step	Hyp	Ref	Expression
1		ltnr 8392	. . . 4 |- ( A e. RR -> -. A < A )
2		breq2 4129	. . . . 5 |- ( B = A -> ( A < B <-> A < A ) )
3	2	notbid 677	. . . 4 |- ( B = A -> ( -. A < B <-> -. A < A ) )
4	1, 3	syl5ibrcom 157	. . 3 |- ( A e. RR -> ( B = A -> -. A < B ) )
5	4	necon2ad 2477	. 2 |- ( A e. RR -> ( A < B -> B =/= A ) )
6	5	imp 124	1 |- ( ( A e. RR /\ A < B ) -> B =/= A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4125   RRcr 8168   < clt 8350

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-pre-ltirr 8281

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-pnf 8352  df-mnf 8353  df-ltxr 8355

This theorem is referenced by:  gtneii  8411  ltnei  8419  gtned  8428  gt0ne0  8745  lt0ne0  8746  gt0ne0d  8830  nngt1ne1  9318  zdceq  9699  qdceq  10657  coprm  12900  phibndlem  12972  tridceq  17011