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Theorem axapti 7977
Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7876 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
axapti  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <  B  \/  B  <  A ) )  ->  A  =  B )

Proof of Theorem axapti
StepHypRef Expression
1 ltxrlt 7972 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
2 ltxrlt 7972 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  B 
<RR  A ) )
32ancoms 266 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  <  A  <->  B 
<RR  A ) )
41, 3orbi12d 788 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  < 
B  \/  B  < 
A )  <->  ( A  <RR  B  \/  B  <RR  A ) ) )
54notbid 662 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  <  B  \/  B  <  A )  <->  -.  ( A  <RR  B  \/  B  <RR  A ) ) )
6 ax-pre-apti 7876 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <RR  B  \/  B  <RR  A ) )  ->  A  =  B )
763expia 1200 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A 
<RR  B  \/  B  <RR  A )  ->  A  =  B ) )
85, 7sylbid 149 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  <  B  \/  B  <  A )  ->  A  =  B ) )
983impia 1195 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <  B  \/  B  <  A ) )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 703    /\ w3a 973    = wceq 1348    e. wcel 2141   class class class wbr 3987   RRcr 7760    <RR cltrr 7765    < clt 7941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-pre-apti 7876
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-xp 4615  df-pnf 7943  df-mnf 7944  df-ltxr 7946
This theorem is referenced by:  lttri3  7986  reapti  8485
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