ixxex - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ixxex		Structured version Visualization version GIF version

Theorem ixxex 13324

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 12953	. . . 4 ⊢ ℝ^* ∈ V
2	1, 1	xpex 7732	. . 3 ⊢ (ℝ^* × ℝ^*) ∈ V
3	1	pwex 5338	. . 3 ⊢ 𝒫 ℝ^* ∈ V
4	2, 3	xpex 7732	. 2 ⊢ ((ℝ^* × ℝ^) × 𝒫 ℝ^) ∈ V
5		ixx.1	. . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
6	5	ixxf 13323	. . 3 ⊢ 𝑂:(ℝ^* × ℝ^)⟶𝒫 ℝ^
7		fssxp 6718	. . 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 5280	1 ⊢ 𝑂 ∈ V

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3408 Vcvv 3450 ⊆ wss 3917 𝒫 cpw 4566 class class class wbr 5110 × cxp 5639 ⟶wf 6510 ∈ cmpo 7392 ℝ^*cxr 11214

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132

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 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-fv 6522 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-xr 11219

This theorem is referenced by: iooex 13336 isbasisrelowl 37353 relowlpssretop 37359

W3C validator