Theorem ltne 8231

Description: 'Less than' implies not equal. See also ltap 8780 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 8223	. . . 4 |- ( A e. RR -> -. A < A )
2		breq2 4087	. . . . 5 |- ( B = A -> ( A < B <-> A < A ) )
3	2	notbid 671	. . . 4 |- ( B = A -> ( -. A < B <-> -. A < A ) )
4	1, 3	syl5ibrcom 157	. . 3 |- ( A e. RR -> ( B = A -> -. A < B ) )
5	4	necon2ad 2457	. 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 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4083   RRcr 7998   < clt 8181

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-pre-ltirr 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-pnf 8183  df-mnf 8184  df-ltxr 8186

This theorem is referenced by:  gtneii  8242  ltnei  8250  gtned  8259  gt0ne0  8574  lt0ne0  8575  gt0ne0d  8659  nngt1ne1  9145  zdceq  9522  qdceq  10464  coprm  12666  phibndlem  12738  tridceq  16424