ixxex - Metamath Proof Explorer

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Theorem ixxex 13373

Description: The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

**Hypothesis**
Ref	Expression
ixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})

**Assertion**
Ref	Expression
ixxex	⊢ 𝑂 ∈ V

Distinct variable groups: 𝑥,𝑦,𝑧,𝑅 𝑥,𝑆,𝑦,𝑧 Allowed substitution hints: 𝑂(𝑥,𝑦,𝑧)

**Proof of Theorem ixxex**
Step	Hyp	Ref	Expression
1		xrex 13003	. . . 4 ⊢ ℝ^* ∈ V
2	1, 1	xpex 7747	. . 3 ⊢ (ℝ^* × ℝ^*) ∈ V
3	1	pwex 5350	. . 3 ⊢ 𝒫 ℝ^* ∈ V
4	2, 3	xpex 7747	. 2 ⊢ ((ℝ^* × ℝ^) × 𝒫 ℝ^) ∈ V
5		ixx.1	. . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
6	5	ixxf 13372	. . 3 ⊢ 𝑂:(ℝ^* × ℝ^)⟶𝒫 ℝ^
7		fssxp 6733	. . 3 ⊢ (𝑂:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → 𝑂 ⊆ ((ℝ^* × ℝ^) × 𝒫 ℝ^))
8	6, 7	ax-mp 5	. 2 ⊢ 𝑂 ⊆ ((ℝ^* × ℝ^) × 𝒫 ℝ^)
9	4, 8	ssexi 5292	1 ⊢ 𝑂 ∈ V

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3415 Vcvv 3459 ⊆ wss 3926 𝒫 cpw 4575 class class class wbr 5119 × cxp 5652 ⟶wf 6527 ∈ cmpo 7407 ℝ^*cxr 11268

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-fv 6539 df-oprab 7409 df-mpo 7410 df-1st 7988 df-2nd 7989 df-xr 11273

This theorem is referenced by: iooex 13385 isbasisrelowl 37376 relowlpssretop 37382

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