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Theorem triap 14061
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 8552 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B # A )
213expia 1200 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> B # A ) )
3 recn 7907 . . . . . 6 |- ( A e. RR -> A e. CC )
4 recn 7907 . . . . . 6 |- ( B e. RR -> B e. CC )
5 apsym 8525 . . . . . 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 707 . . . . 5 |- ( A # B -> ( A # B \/ -. A # B ) )
9 df-dc 830 . . . . 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 8541 . . . . 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 706 . . . . 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 8552 . . . . . 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 1200 . . . 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 1299 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ A = B \/ B < A ) -> DECID A # B ) )
22 reaplt 8507 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
23 orc 707 . . . . . . 7 |- ( A < B -> ( A < B \/ A = B ) )
2423orim1i 755 . . . . . 6 |- ( ( A < B \/ B < A ) -> ( ( A < B \/ A = B ) \/ B < A ) )
25 df-3or 974 . . . . . 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 1162 . . . . 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 712 . . 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 703  DECID wdc 829   \/ w3o 972   /\ w3a 973   = wceq 1348   e. wcel 2141   class class class wbr 3989   CCcc 7772   RRcr 7773   < clt 7954   # cap 8500
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 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  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-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501
This theorem is referenced by:  reap0  14090
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