Theorem gtneii 8242

Description: 'Less than' implies not equal. See also gtapii 8781 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)

Hypotheses

Ref	Expression
lt.1	|- A e. RR
ltneii.2	|- A < B

Assertion

Ref	Expression
gtneii	|- B =/= A

Proof of Theorem gtneii

Step	Hyp	Ref	Expression
1		lt.1	. 2 |- A e. RR
2		ltneii.2	. 2 |- A < B
3		ltne 8231	. 2 |- ( ( A e. RR /\ A < B ) -> B =/= A )
4	1, 2, 3	mp2an 426	1 |- B =/= A

Syntax hints:   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:  ltneii  8243  ine0  8540  fztpval  10279  ene1  12296  3lcm2e6  12682  starvndxnbasendx  13175  starvndxnplusgndx  13176  starvndxnmulrndx  13177  scandxnbasendx  13187  scandxnplusgndx  13188  scandxnmulrndx  13189  vscandxnbasendx  13192  vscandxnplusgndx  13193  vscandxnmulrndx  13194  vscandxnscandx  13195  ipndxnbasendx  13205  ipndxnplusgndx  13206  ipndxnmulrndx  13207  tsetndxnbasendx  13224  tsetndxnplusgndx  13225  tsetndxnmulrndx  13226  tsetndxnstarvndx  13227  slotstnscsi  13228  plendxnbasendx  13238  plendxnplusgndx  13239  plendxnmulrndx  13240  plendxnscandx  13241  plendxnvscandx  13242  dsndxnbasendx  13253  dsndxnplusgndx  13254  dsndxnmulrndx  13255  slotsdnscsi  13256  dsndxntsetndx  13257  unifndxnbasendx  13263  unifndxntsetndx  13264  setsmsdsg  15154  2logb9irr  15645  2logb3irr  15647  2logb9irrap  15651