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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > icomnfinre | Structured version Visualization version GIF version |
Description: A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
icomnfinre.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
Ref | Expression |
---|---|
icomnfinre | ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 11213 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → -∞ ∈ ℝ*) |
3 | icomnfinre.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | 3 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝐴 ∈ ℝ*) |
5 | elinel2 4157 | . . . . 5 ⊢ (𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ) → 𝑥 ∈ ℝ) | |
6 | 5 | adantl 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ ℝ) |
7 | 6 | mnfltd 13046 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → -∞ < 𝑥) |
8 | elinel1 4156 | . . . . . 6 ⊢ (𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ) → 𝑥 ∈ (-∞[,)𝐴)) | |
9 | 8 | adantl 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ (-∞[,)𝐴)) |
10 | 2, 4, 9 | icoltubd 43790 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 < 𝐴) |
11 | 2, 4, 6, 7, 10 | eliood 43743 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ (-∞(,)𝐴)) |
12 | 11 | ssd 43297 | . 2 ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) ⊆ (-∞(,)𝐴)) |
13 | ioossico 13356 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ (-∞[,)𝐴) | |
14 | ioossre 13326 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ ℝ | |
15 | 13, 14 | ssini 4192 | . . 3 ⊢ (-∞(,)𝐴) ⊆ ((-∞[,)𝐴) ∩ ℝ) |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → (-∞(,)𝐴) ⊆ ((-∞[,)𝐴) ∩ ℝ)) |
17 | 12, 16 | eqssd 3962 | 1 ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3910 ⊆ wss 3911 (class class class)co 7358 ℝcr 11051 -∞cmnf 11188 ℝ*cxr 11189 (,)cioo 13265 [,)cico 13267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-ioo 13269 df-ico 13271 |
This theorem is referenced by: preimaioomnf 44967 |
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