elioo5 - Metamath Proof Explorer

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Theorem elioo5 13574

Description: Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)

**Assertion**
Ref	Expression
elioo5	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))

**Proof of Theorem elioo5**
Step	Hyp	Ref	Expression
1		elioo1 13556	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
2	1	3adant3 1150	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
3		3anass 1111	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (𝐶 ∈ ℝ^* ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
4	3	baibr 546	. . 3 ⊢ (𝐶 ∈ ℝ^* → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
5	4	3ad2ant3 1153	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
6	2, 5	bitr4d 285	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 ∈ wcel 2147 class class class wbr 5108 (class class class)co 7451 ℝ^*cxr 11381 < clt 11382 (,)cioo 13516

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-iota 6520 df-fun 6568 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-xr 11386 df-ioo 13520

This theorem is used by: iooshf 13597 iooneg 13642 lhop1 26389 tan2h 38803 poimir 38839 ftc1anclem1 38879 dvrelog2b 43382 aks4d1p1p6 43389 aks4d1p1p5 43391

W3C validator