xrltmaxsup - Intuitionistic Logic Explorer

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Theorem xrltmaxsup 11840

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 1026	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ^, < )) → 𝐴 ∈ ℝ^*)
2		simpl2 1027	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ^, < )) → 𝐵 ∈ ℝ^*)
3		simpl3 1028	. . 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 11839	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ^*, < ))) → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))
6	1, 2, 3, 4, 5	syl22anc 1274	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ^, < )) → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))
7		simpl3 1028	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐴) → 𝐶 ∈ ℝ^)
8		simpl1 1026	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐴) → 𝐴 ∈ ℝ^)
9		simpl2 1027	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐴) → 𝐵 ∈ ℝ^)
10		xrmaxcl 11835	. . . . 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 11836	. . . . 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 10051	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐴) → 𝐶 < sup({𝐴, 𝐵}, ℝ^, < ))
16		simpl3 1028	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐵) → 𝐶 ∈ ℝ^)
17		simpl2 1027	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐵) → 𝐵 ∈ ℝ^)
18		simpl1 1026	. . . . 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 11837	. . . . 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 10051	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐶 < 𝐵) → 𝐶 < sup({𝐴, 𝐵}, ℝ^, < ))
24	15, 23	jaodan 804	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)) → 𝐶 < sup({𝐴, 𝐵}, ℝ^, < ))
25	6, 24	impbida 600	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 715 ∧ w3a 1004 ∈ wcel 2201 {cpr 3671 class class class wbr 4089 supcsup 7186 ℝ^*cxr 8218 < clt 8219 ≤ cle 8220

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 2203 ax-14 2204 ax-ext 2212 ax-coll 4205 ax-sep 4208 ax-nul 4216 ax-pow 4266 ax-pr 4301 ax-un 4532 ax-setind 4637 ax-iinf 4688 ax-cnex 8128 ax-resscn 8129 ax-1cn 8130 ax-1re 8131 ax-icn 8132 ax-addcl 8133 ax-addrcl 8134 ax-mulcl 8135 ax-mulrcl 8136 ax-addcom 8137 ax-mulcom 8138 ax-addass 8139 ax-mulass 8140 ax-distr 8141 ax-i2m1 8142 ax-0lt1 8143 ax-1rid 8144 ax-0id 8145 ax-rnegex 8146 ax-precex 8147 ax-cnre 8148 ax-pre-ltirr 8149 ax-pre-ltwlin 8150 ax-pre-lttrn 8151 ax-pre-apti 8152 ax-pre-ltadd 8153 ax-pre-mulgt0 8154 ax-pre-mulext 8155 ax-arch 8156 ax-caucvg 8157

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 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ne 2402 df-nel 2497 df-ral 2514 df-rex 2515 df-reu 2516 df-rmo 2517 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-nul 3494 df-if 3605 df-pw 3655 df-sn 3676 df-pr 3677 df-op 3679 df-uni 3895 df-int 3930 df-iun 3973 df-br 4090 df-opab 4152 df-mpt 4153 df-tr 4189 df-id 4392 df-po 4395 df-iso 4396 df-iord 4465 df-on 4467 df-ilim 4468 df-suc 4470 df-iom 4691 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-res 4739 df-ima 4740 df-iota 5288 df-fun 5330 df-fn 5331 df-f 5332 df-f1 5333 df-fo 5334 df-f1o 5335 df-fv 5336 df-riota 5976 df-ov 6026 df-oprab 6027 df-mpo 6028 df-1st 6308 df-2nd 6309 df-recs 6476 df-frec 6562 df-sup 7188 df-pnf 8221 df-mnf 8222 df-xr 8223 df-ltxr 8224 df-le 8225 df-sub 8357 df-neg 8358 df-reap 8760 df-ap 8767 df-div 8858 df-inn 9149 df-2 9207 df-3 9208 df-4 9209 df-n0 9408 df-z 9485 df-uz 9761 df-rp 9894 df-seqfrec 10716 df-exp 10807 df-cj 11425 df-re 11426 df-im 11427 df-rsqrt 11581 df-abs 11582

This theorem is referenced by: xrmaxlesup 11842 xrmaxaddlem 11843 xrmaxadd 11844 xrminltinf 11855

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