xrgtnelicc - Mathbox for Glauco Siliprandi

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Theorem xrgtnelicc 46119

Description: A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

**Hypotheses**
Ref	Expression
xrgtnelicc.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xrgtnelicc.2	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
xrgtnelicc.3	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
xrgtnelicc.4	⊢ (𝜑 → 𝐵 < 𝐶)

**Assertion**
Ref	Expression
xrgtnelicc	⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

**Proof of Theorem xrgtnelicc**
Step	Hyp	Ref	Expression
1		xrgtnelicc.4	. . . 4 ⊢ (𝜑 → 𝐵 < 𝐶)
2		xrgtnelicc.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
3		xrgtnelicc.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
4		xrltnle 11251	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
5	2, 3, 4	syl2anc 593	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
6	1, 5	mpbid 234	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵)
7	6	intnand 492	. 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
8		xrgtnelicc.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
9		elicc4 13419	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
10	8, 2, 3, 9	syl3anc 1392	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	7, 10	mtbird 327	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∈ wcel 2144 class class class wbr 5102 (class class class)co 7398 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219 [,]cicc 13354

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-sep 5248 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ral 3079 df-rex 3089 df-rab 3417 df-v 3458 df-sbc 3747 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5103 df-opab 5165 df-id 5544 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-iota 6479 df-fun 6525 df-fv 6531 df-ov 7401 df-oprab 7402 df-mpo 7403 df-xr 11222 df-le 11224 df-icc 13358

This theorem is referenced by: iccdificc 46120

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