ioorebasg - Intuitionistic Logic Explorer

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Theorem ioorebasg 8773

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 8769	. . 3 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		ffn 5007	. . 3 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
3	1, 2	ax-mp 7	. 2 ⊢ (,) Fn (ℝ^* × ℝ^*)
4		fnovrn 5609	. 2 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ 𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ ran (,))
5	3, 4	mp3an1 1219	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ ran (,))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 𝒫 cpw 3356 × cxp 4304 ran crn 4307 Fn wfn 4858 ⟶wf 4859 (class class class)co 5473 ℝcr 6831 ℝ^*cxr 7001 (,)cioo 8686

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3871 ax-pow 3923 ax-pr 3940 ax-un 4141 ax-setind 4230 ax-cnex 6918 ax-resscn 6919 ax-pre-ltirr 6939 ax-pre-ltwlin 6940 ax-pre-lttrn 6941

This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2308 df-rex 2309 df-rab 2312 df-v 2556 df-sbc 2762 df-csb 2850 df-dif 2917 df-un 2919 df-in 2921 df-ss 2928 df-pw 3358 df-sn 3378 df-pr 3379 df-op 3381 df-uni 3577 df-iun 3655 df-br 3761 df-opab 3815 df-mpt 3816 df-id 4026 df-po 4029 df-iso 4030 df-xp 4312 df-rel 4313 df-cnv 4314 df-co 4315 df-dm 4316 df-rn 4317 df-res 4318 df-ima 4319 df-iota 4828 df-fun 4865 df-fn 4866 df-f 4867 df-fv 4871 df-ov 5476 df-oprab 5477 df-mpt2 5478 df-1st 5728 df-2nd 5729 df-pnf 7004 df-mnf 7005 df-xr 7006 df-ltxr 7007 df-le 7008 df-ioo 8690

This theorem is referenced by: (None)

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