elioo5 - Intuitionistic Logic Explorer

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Theorem elioo5 10085

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 10063	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
2	1	3adant3 1020	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
3		3anass 985	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (𝐶 ∈ ℝ^* ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
4	3	baibr 922	. . 3 ⊢ (𝐶 ∈ ℝ^* → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
5	4	3ad2ant3 1023	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
6	2, 5	bitr4d 191	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 981 ∈ wcel 2177 class class class wbr 4054 (class class class)co 5962 ℝ^*cxr 8136 < clt 8137 (,)cioo 10040

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4173 ax-pow 4229 ax-pr 4264 ax-un 4493 ax-setind 4598 ax-cnex 8046 ax-resscn 8047

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-ral 2490 df-rex 2491 df-rab 2494 df-v 2775 df-sbc 3003 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3860 df-br 4055 df-opab 4117 df-id 4353 df-xp 4694 df-rel 4695 df-cnv 4696 df-co 4697 df-dm 4698 df-iota 5246 df-fun 5287 df-fv 5293 df-ov 5965 df-oprab 5966 df-mpo 5967 df-pnf 8139 df-mnf 8140 df-xr 8141 df-ioo 10044

This theorem is referenced by: iooshf 10104 iooneg 10140

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