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Theorem triap 16633
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 8812 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B # A )
213expia 1231 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> B # A ) )
3 recn 8164 . . . . . 6 |- ( A e. RR -> A e. CC )
4 recn 8164 . . . . . 6 |- ( B e. RR -> B e. CC )
5 apsym 8785 . . . . . 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 719 . . . . 5 |- ( A # B -> ( A # B \/ -. A # B ) )
9 df-dc 842 . . . . 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 8801 . . . . 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 718 . . . . 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 8812 . . . . . 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 1231 . . . 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 1340 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ A = B \/ B < A ) -> DECID A # B ) )
22 reaplt 8767 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
23 orc 719 . . . . . . 7 |- ( A < B -> ( A < B \/ A = B ) )
2423orim1i 767 . . . . . 6 |- ( ( A < B \/ B < A ) -> ( ( A < B \/ A = B ) \/ B < A ) )
25 df-3or 1005 . . . . . 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 1193 . . . . 5 |- ( A = B -> ( A < B \/ A = B \/ B < A ) )
2913, 28biimtrrdi 164 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( -. A # B -> ( A < B \/ A = B \/ B < A ) ) )
3027, 29jaod 724 . . 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 715  DECID wdc 841   \/ w3o 1003   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088   CCcc 8029   RRcr 8030   < clt 8213   # cap 8760
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148
This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761
This theorem is referenced by:  reap0  16662  cndcap  16663
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