Theorem ltne 8358

Description: 'Less than' implies not equal. See also ltap 8907 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 8350	. . . 4 |- ( A e. RR -> -. A < A )
2		breq2 4113	. . . . 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 2469	. 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 2203   =/= wne 2412   class class class wbr 4109   RRcr 8126   < clt 8308

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-pre-ltirr 8239

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-pnf 8310  df-mnf 8311  df-ltxr 8313

This theorem is referenced by:  gtneii  8369  ltnei  8377  gtned  8386  gt0ne0  8701  lt0ne0  8702  gt0ne0d  8786  nngt1ne1  9272  zdceq  9653  qdceq  10604  coprm  12841  phibndlem  12913  tridceq  16841