xrltmaxsup - Intuitionistic Logic Explorer

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Theorem xrltmaxsup 11231

Description: The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.)

**Assertion**
Ref	Expression
xrltmaxsup	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐶 < sup({𝐴, 𝐵}, ℝ^, < ) ↔ (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)))

**Proof of Theorem xrltmaxsup**
Step	Hyp	Ref	Expression
1		simpl1 1000	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ^, < )) → 𝐴 ∈ ℝ^*)
2		simpl2 1001	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ^, < )) → 𝐵 ∈ ℝ^*)
3		simpl3 1002	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ^, < )) → 𝐶 ∈ ℝ^*)
4		simpr 110	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ^, < )) → 𝐶 < sup({𝐴, 𝐵}, ℝ^*, < ))
5		xrmaxleastlt 11230	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ^*, < ))) → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))
6	1, 2, 3, 4, 5	syl22anc 1239	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ^, < )) → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))
7		simpl3 1002	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐴) → 𝐶 ∈ ℝ^)
8		simpl1 1000	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐴) → 𝐴 ∈ ℝ^)
9		simpl2 1001	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐴) → 𝐵 ∈ ℝ^)
10		xrmaxcl 11226	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → sup({𝐴, 𝐵}, ℝ^, < ) ∈ ℝ^*)
11	8, 9, 10	syl2anc 411	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐴) → sup({𝐴, 𝐵}, ℝ^, < ) ∈ ℝ^*)
12		simpr 110	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐶 < 𝐴) → 𝐶 < 𝐴)
13		xrmax1sup 11227	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ≤ sup({𝐴, 𝐵}, ℝ^, < ))
14	8, 9, 13	syl2anc 411	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐴) → 𝐴 ≤ sup({𝐴, 𝐵}, ℝ^, < ))
15	7, 8, 11, 12, 14	xrltletrd 9780	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐴) → 𝐶 < sup({𝐴, 𝐵}, ℝ^, < ))
16		simpl3 1002	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐵) → 𝐶 ∈ ℝ^)
17		simpl2 1001	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐵) → 𝐵 ∈ ℝ^)
18		simpl1 1000	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐵) → 𝐴 ∈ ℝ^)
19	18, 17, 10	syl2anc 411	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐵) → sup({𝐴, 𝐵}, ℝ^, < ) ∈ ℝ^*)
20		simpr 110	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐶 < 𝐵) → 𝐶 < 𝐵)
21		xrmax2sup 11228	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐵 ≤ sup({𝐴, 𝐵}, ℝ^, < ))
22	18, 17, 21	syl2anc 411	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐵) → 𝐵 ≤ sup({𝐴, 𝐵}, ℝ^, < ))
23	16, 17, 19, 20, 22	xrltletrd 9780	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐵) → 𝐶 < sup({𝐴, 𝐵}, ℝ^, < ))
24	15, 23	jaodan 797	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)) → 𝐶 < sup({𝐴, 𝐵}, ℝ^, < ))
25	6, 24	impbida 596	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐶 < sup({𝐴, 𝐵}, ℝ^, < ) ↔ (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 ∧ w3a 978 ∈ wcel 2146 {cpr 3590 class class class wbr 3998 supcsup 6971 ℝ^*cxr 7965 < clt 7966 ≤ cle 7967

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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-sup 6973 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8602 df-inn 8891 df-2 8949 df-3 8950 df-4 8951 df-n0 9148 df-z 9225 df-uz 9500 df-rp 9623 df-seqfrec 10414 df-exp 10488 df-cj 10817 df-re 10818 df-im 10819 df-rsqrt 10973 df-abs 10974

This theorem is referenced by: xrmaxlesup 11233 xrmaxaddlem 11234 xrmaxadd 11235 xrminltinf 11246

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