ioorebasg - Intuitionistic Logic Explorer

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Theorem ioorebasg 9151

Description: Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)

**Assertion**
Ref	Expression
ioorebasg	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ ran (,))

**Proof of Theorem ioorebasg**
Step	Hyp	Ref	Expression
1		ioof 9147	. . 3 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		ffn 5098	. . 3 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
3	1, 2	ax-mp 7	. 2 ⊢ (,) Fn (ℝ^* × ℝ^*)
4		fnovrn 5701	. 2 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ 𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ ran (,))
5	3, 4	mp3an1 1256	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ ran (,))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1434 𝒫 cpw 3401 × cxp 4390 ran crn 4393 Fn wfn 4948 ⟶wf 4949 (class class class)co 5565 ℝcr 7119 ℝ^*cxr 7291 (,)cioo 9064

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-cnex 7206 ax-resscn 7207 ax-pre-ltirr 7227 ax-pre-ltwlin 7228 ax-pre-lttrn 7229

This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2613 df-sbc 2826 df-csb 2919 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-iun 3701 df-br 3807 df-opab 3861 df-mpt 3862 df-id 4077 df-po 4080 df-iso 4081 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-fv 4961 df-ov 5568 df-oprab 5569 df-mpt2 5570 df-1st 5820 df-2nd 5821 df-pnf 7294 df-mnf 7295 df-xr 7296 df-ltxr 7297 df-le 7298 df-ioo 9068

This theorem is referenced by: (None)

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