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Theorem triap 15131
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 8604 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B # A )
213expia 1206 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> B # A ) )
3 recn 7958 . . . . . 6 |- ( A e. RR -> A e. CC )
4 recn 7958 . . . . . 6 |- ( B e. RR -> B e. CC )
5 apsym 8577 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> B # A ) )
63, 4, 5syl2an 289 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> B # A ) )
72, 6sylibrd 169 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A # B ) )
8 orc 713 . . . . 5 |- ( A # B -> ( A # B \/ -. A # B ) )
9 df-dc 836 . . . . 5 |- (DECID A # B <-> ( A # B \/ -. A # B ) )
108, 9sylibr 134 . . . 4 |- ( A # B -> DECID A # B )
117, 10syl6 33 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> DECID A # B ) )
12 apti 8593 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> -. A # B ) )
133, 4, 12syl2an 289 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A # B ) )
14 olc 712 . . . . 5 |- ( -. A # B -> ( A # B \/ -. A # B ) )
1514, 9sylibr 134 . . . 4 |- ( -. A # B -> DECID A # B )
1613, 15biimtrdi 163 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A = B -> DECID A # B ) )
17 ltap 8604 . . . . . 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 1206 . . . 4 |- ( ( B e. RR /\ A e. RR ) -> ( B < A -> DECID A # B ) )
2019ancoms 268 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( B < A -> DECID A # B ) )
2111, 16, 203jaod 1314 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ A = B \/ B < A ) -> DECID A # B ) )
22 reaplt 8559 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
23 orc 713 . . . . . . 7 |- ( A < B -> ( A < B \/ A = B ) )
2423orim1i 761 . . . . . 6 |- ( ( A < B \/ B < A ) -> ( ( A < B \/ A = B ) \/ B < A ) )
25 df-3or 980 . . . . . 6 |- ( ( A < B \/ A = B \/ B < A ) <-> ( ( A < B \/ A = B ) \/ B < A ) )
2624, 25sylibr 134 . . . . 5 |- ( ( A < B \/ B < A ) -> ( A < B \/ A = B \/ B < A ) )
2722, 26biimtrdi 163 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A # B -> ( A < B \/ A = B \/ B < A ) ) )
28 3mix2 1168 . . . . 5 |- ( A = B -> ( A < B \/ A = B \/ B < A ) )
2913, 28syl6bir 164 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -. A # B -> ( A < B \/ A = B \/ B < A ) ) )
3027, 29jaod 718 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( A # B \/ -. A # B ) -> ( A < B \/ A = B \/ B < A ) ) )
319, 30biimtrid 152 . 2 |- ( ( A e. RR /\ B e. RR ) -> (DECID A # B -> ( A < B \/ A = B \/ B < A ) ) )
3221, 31impbid 129 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 104   <-> wb 105   \/ wo 709  DECID wdc 835   \/ w3o 978   /\ w3a 979   = wceq 1363   e. wcel 2158   class class class wbr 4015   CCcc 7823   RRcr 7824   < clt 8006   # cap 8552
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553
This theorem is referenced by:  reap0  15160  cndcap  15161
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