Theorem ltne 8157

Description: 'Less than' implies not equal. See also ltap 8706 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 8149	. . . 4 |- ( A e. RR -> -. A < A )
2		breq2 4048	. . . . 5 |- ( B = A -> ( A < B <-> A < A ) )
3	2	notbid 669	. . . 4 |- ( B = A -> ( -. A < B <-> -. A < A ) )
4	1, 3	syl5ibrcom 157	. . 3 |- ( A e. RR -> ( B = A -> -. A < B ) )
5	4	necon2ad 2433	. 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 1373   e. wcel 2176   =/= wne 2376   class class class wbr 4044   RRcr 7924   < clt 8107

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-pre-ltirr 8037

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-pnf 8109  df-mnf 8110  df-ltxr 8112

This theorem is referenced by:  gtneii  8168  ltnei  8176  gtned  8185  gt0ne0  8500  lt0ne0  8501  gt0ne0d  8585  nngt1ne1  9071  zdceq  9448  qdceq  10387  coprm  12466  phibndlem  12538  tridceq  15999