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Theorem triap 13600
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 8508 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B # A )
213expia 1187 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> B # A ) )
3 recn 7865 . . . . . 6 |- ( A e. RR -> A e. CC )
4 recn 7865 . . . . . 6 |- ( B e. RR -> B e. CC )
5 apsym 8481 . . . . . 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 821 . . . . 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 8497 . . . . 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 8508 . . . . . 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 1187 . . . 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 1286 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ A = B \/ B < A ) -> DECID A # B ) )
22 reaplt 8463 . . . . 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 964 . . . . . 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 1152 . . . . 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 820   \/ w3o 962   /\ w3a 963   = wceq 1335   e. wcel 2128   class class class wbr 3965   CCcc 7730   RRcr 7731   < clt 7912   # cap 8456
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-ltxr 7917  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457
This theorem is referenced by:  reap0  13629
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