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Theorem leordtvallem1 23378

Description: Lemma for leordtval 23381. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Hypothesis**
Ref	Expression
leordtval.1	⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))

**Assertion**
Ref	Expression
leordtvallem1	⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})

Distinct variable group: 𝑥,𝑦 Allowed substitution hints: 𝐴(𝑥, 𝑦)

**Proof of Theorem leordtvallem1**
Step	Hyp	Ref	Expression
1		leordtval.1	. 2 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
2		iocssxr 13464	. . . . . 6 ⊢ (𝑥(,]+∞) ⊆ ℝ^*
3		sseqin2 4175	. . . . . 6 ⊢ ((𝑥(,]+∞) ⊆ ℝ^* ↔ (ℝ^* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞))
4	2, 3	mpbi 233	. . . . 5 ⊢ (ℝ^* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞)
5		simpl 487	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑥 ∈ ℝ^)
6		pnfxr 11269	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
7		elioc1 13420	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ^ ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
8	5, 6, 7	sylancl 597	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ^ ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
9		simpr 489	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑦 ∈ ℝ^)
10		pnfge 13161	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
11	9, 10	jccir 530	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ ℝ^ ∧ 𝑦 ≤ +∞))
12	11	biantrurd 541	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑥 < 𝑦 ↔ ((𝑦 ∈ ℝ^ ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦)))
13		3anan32 1112	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞) ↔ ((𝑦 ∈ ℝ^* ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦))
14	12, 13	bitr4di 292	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑥 < 𝑦 ↔ (𝑦 ∈ ℝ^ ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
15		xrltnle 11282	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
16	8, 14, 15	3bitr2d 310	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 ∈ (𝑥(,]+∞) ↔ ¬ 𝑦 ≤ 𝑥))
17	16	rabbi2dva 4177	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (ℝ^* ∩ (𝑥(,]+∞)) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
18	4, 17	eqtr3id 2811	. . . 4 ⊢ (𝑥 ∈ ℝ^* → (𝑥(,]+∞) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
19	18	mpteq2ia 5205	. . 3 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
20	19	rneqi 5926	. 2 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
21	1, 20	eqtri 2785	1 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 {crab 3415 ∩ cin 3903 ⊆ wss 3904 class class class wbr 5108 ↦ cmpt 5191 ran crn 5661 (class class class)co 7412 +∞cpnf 11246 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 (,]cioc 13379

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7984 df-2nd 7985 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-ioc 13383

This theorem is used by: leordtval2 23380 leordtval 23381

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