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Theorem triap 13908
Description: Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.)
Assertion
Ref Expression
triap |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ A = B \/ B < A ) <-> DECID A # B ) )
Proof of Theorem triap
StepHypRef Expression
1 ltap 8531 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B # A )
213expia 1195 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> B # A ) )
3 recn 7886 . . . . . 6 |- ( A e. RR -> A e. CC )
4 recn 7886 . . . . . 6 |- ( B e. RR -> B e. CC )
5 apsym 8504 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> B # A ) )
63, 4, 5syl2an 287 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> B # A ) )
72, 6sylibrd 168 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A # B ) )
8 orc 702 . . . . 5 |- ( A # B -> ( A # B \/ -. A # B ) )
9 df-dc 825 . . . . 5 |- (DECID A # B <-> ( A # B \/ -. A # B ) )
108, 9sylibr 133 . . . 4 |- ( A # B -> DECID A # B )
117, 10syl6 33 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> DECID A # B ) )
12 apti 8520 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> -. A # B ) )
133, 4, 12syl2an 287 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A # B ) )
14 olc 701 . . . . 5 |- ( -. A # B -> ( A # B \/ -. A # B ) )
1514, 9sylibr 133 . . . 4 |- ( -. A # B -> DECID A # B )
1613, 15syl6bi 162 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A = B -> DECID A # B ) )
17 ltap 8531 . . . . . 6 |- ( ( B e. RR /\ A e. RR /\ B < A ) -> A # B )
1817, 10syl 14 . . . . 5 |- ( ( B e. RR /\ A e. RR /\ B < A ) -> DECID A # B )
19183expia 1195 . . . 4 |- ( ( B e. RR /\ A e. RR ) -> ( B < A -> DECID A # B ) )
2019ancoms 266 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( B < A -> DECID A # B ) )
2111, 16, 203jaod 1294 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ A = B \/ B < A ) -> DECID A # B ) )
22 reaplt 8486 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
23 orc 702 . . . . . . 7 |- ( A < B -> ( A < B \/ A = B ) )
2423orim1i 750 . . . . . 6 |- ( ( A < B \/ B < A ) -> ( ( A < B \/ A = B ) \/ B < A ) )
25 df-3or 969 . . . . . 6 |- ( ( A < B \/ A = B \/ B < A ) <-> ( ( A < B \/ A = B ) \/ B < A ) )
2624, 25sylibr 133 . . . . 5 |- ( ( A < B \/ B < A ) -> ( A < B \/ A = B \/ B < A ) )
2722, 26syl6bi 162 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A # B -> ( A < B \/ A = B \/ B < A ) ) )
28 3mix2 1157 . . . . 5 |- ( A = B -> ( A < B \/ A = B \/ B < A ) )
2913, 28syl6bir 163 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -. A # B -> ( A < B \/ A = B \/ B < A ) ) )
3027, 29jaod 707 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( A # B \/ -. A # B ) -> ( A < B \/ A = B \/ B < A ) ) )
319, 30syl5bi 151 . 2 |- ( ( A e. RR /\ B e. RR ) -> (DECID A # B -> ( A < B \/ A = B \/ B < A ) ) )
3221, 31impbid 128 1 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ A = B \/ B < A ) <-> DECID A # B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   \/ w3o 967   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982   CCcc 7751   RRcr 7752   < clt 7933   # cap 8479
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-ltxr 7938  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480
This theorem is referenced by:  reap0  13937
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