icoub - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > icoub		Structured version Visualization version GIF version

Theorem icoub 45472

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 13453	. . . . 5 ⊢ (𝐴[,)𝐵) ⊆ ℝ^*
3		id 22	. . . . 5 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → 𝐵 ∈ (𝐴[,)𝐵))
4	2, 3	sselid 3961	. . . 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 45454	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 < 𝐵)
8	1, 5, 6, 7	syl3anc 1372	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 < 𝐵)
9		xrltnr 13142	. . . 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 2107 class class class wbr 5123 (class class class)co 7412 ℝ^*cxr 11275 < clt 11276 [,)cico 13370

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-pre-lttri 11210 ax-pre-lttrn 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-po 5572 df-so 5573 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6493 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7995 df-2nd 7996 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-ico 13374

This theorem is referenced by: fge0npnf 46315