icoub - Mathbox for Glauco Siliprandi

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Theorem icoub 45497

Description: A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Assertion**
Ref	Expression
icoub	⊢ (𝐴 ∈ ℝ^* → ¬ 𝐵 ∈ (𝐴[,)𝐵))

**Proof of Theorem icoub**
Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ^*)
2		icossxr 13406	. . . . 5 ⊢ (𝐴[,)𝐵) ⊆ ℝ^*
3		id 22	. . . . 5 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → 𝐵 ∈ (𝐴[,)𝐵))
4	2, 3	sselid 3952	. . . 4 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ^*)
5	4	adantl 481	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ^*)
6		simpr 484	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ (𝐴[,)𝐵))
7		icoltub 45479	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 < 𝐵)
8	1, 5, 6, 7	syl3anc 1373	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 < 𝐵)
9		xrltnr 13092	. . . 4 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < 𝐵)
10	4, 9	syl 17	. . 3 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → ¬ 𝐵 < 𝐵)
11	10	adantl 481	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → ¬ 𝐵 < 𝐵)
12	8, 11	pm2.65da 816	1 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐵 ∈ (𝐴[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2109 class class class wbr 5115 (class class class)co 7394 ℝ^*cxr 11225 < clt 11226 [,)cico 13321

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718 ax-cnex 11142 ax-resscn 11143 ax-pre-lttri 11160 ax-pre-lttrn 11161

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-nel 3032 df-ral 3047 df-rex 3056 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-op 4604 df-uni 4880 df-iun 4965 df-br 5116 df-opab 5178 df-mpt 5197 df-id 5541 df-po 5554 df-so 5555 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-ov 7397 df-oprab 7398 df-mpo 7399 df-1st 7977 df-2nd 7978 df-er 8682 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11228 df-mnf 11229 df-xr 11230 df-ltxr 11231 df-ico 13325

This theorem is referenced by: fge0npnf 46338