Theorem ltne 7103

Description: 'Less than' implies not equal. See also ltap 7622 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 7095	. . . 4 |- ( A e. RR -> -. A < A )
2		breq2 3768	. . . . 5 |- ( B = A -> ( A < B <-> A < A ) )
3	2	notbid 592	. . . 4 |- ( B = A -> ( -. A < B <-> -. A < A ) )
4	1, 3	syl5ibrcom 146	. . 3 |- ( A e. RR -> ( B = A -> -. A < B ) )
5	4	necon2ad 2262	. 2 |- ( A e. RR -> ( A < B -> B =/= A ) )
6	5	imp 115	1 |- ( ( A e. RR /\ A < B ) -> B =/= A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   =/= wne 2204   class class class wbr 3764   RRcr 6888   < clt 7060

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-ltxr 7065

This theorem is referenced by:  gtneii  7113  ltnei  7121  gtned  7130  gt0ne0  7422  lt0ne0  7423  gt0ne0d  7504  nngt1ne1  7948  zdceq  8316  qdceq  9102