Theorem ltne 8374

Description: 'Less than' implies not equal. See also ltap 8924 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 8366	. . . 4 |- ( A e. RR -> -. A < A )
2		breq2 4118	. . . . 5 |- ( B = A -> ( A < B <-> A < A ) )
3	2	notbid 673	. . . 4 |- ( B = A -> ( -. A < B <-> -. A < A ) )
4	1, 3	syl5ibrcom 157	. . 3 |- ( A e. RR -> ( B = A -> -. A < B ) )
5	4	necon2ad 2471	. 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 1398   e. wcel 2205   =/= wne 2414   class class class wbr 4114   RRcr 8142   < clt 8324

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-pre-ltirr 8255

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-pnf 8326  df-mnf 8327  df-ltxr 8329

This theorem is referenced by:  gtneii  8385  ltnei  8393  gtned  8402  gt0ne0  8718  lt0ne0  8719  gt0ne0d  8803  nngt1ne1  9289  zdceq  9670  qdceq  10628  coprm  12866  phibndlem  12938  tridceq  16967