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Theorem elicc4 13381

Description: Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)

**Assertion**
Ref	Expression
elicc4	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))

**Proof of Theorem elicc4**
Step	Hyp	Ref	Expression
1		elicc1 13357	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2		3anass 1094	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) ↔ (𝐶 ∈ ℝ^* ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
3	1, 2	bitrdi 287	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))))
4	3	baibd 539	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
5	4	3impa 1109	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2109 class class class wbr 5110 (class class class)co 7390 ℝ^*cxr 11214 ≤ cle 11216 [,]cicc 13316

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 5254 ax-nul 5264 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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 2879 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-iota 6467 df-fun 6516 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-xr 11219 df-icc 13320

This theorem is referenced by: elicc4abs 15293 xrge0addass 32964 esumle 34055 esumlef 34059 sin2h 37611 cos2h 37612 tan2h 37613 ltnelicc 45502 gtnelicc 45505 eliccxrd 45532 xrgtnelicc 45543 limciccioolb 45626 fourierdlem1 46113

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