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Mirrors > Home > MPE Home > Th. List > reltxrnmnf | Structured version Visualization version GIF version |
Description: For all extended real numbers not being minus infinity there is a smaller real number. (Contributed by AV, 5-Sep-2020.) |
Ref | Expression |
---|---|
reltxrnmnf | ⊢ ∀𝑥 ∈ ℝ* (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 12499 | . . 3 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) | |
2 | reltre 12721 | . . . . . 6 ⊢ ∀𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑦 < 𝑥 | |
3 | 2 | rspec 3204 | . . . . 5 ⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ 𝑦 < 𝑥) |
4 | 3 | a1d 25 | . . . 4 ⊢ (𝑥 ∈ ℝ → (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) |
5 | 0red 10632 | . . . . . 6 ⊢ (𝑥 = +∞ → 0 ∈ ℝ) | |
6 | breq1 5060 | . . . . . . 7 ⊢ (𝑦 = 0 → (𝑦 < 𝑥 ↔ 0 < 𝑥)) | |
7 | 6 | adantl 482 | . . . . . 6 ⊢ ((𝑥 = +∞ ∧ 𝑦 = 0) → (𝑦 < 𝑥 ↔ 0 < 𝑥)) |
8 | 0ltpnf 12505 | . . . . . . 7 ⊢ 0 < +∞ | |
9 | breq2 5061 | . . . . . . 7 ⊢ (𝑥 = +∞ → (0 < 𝑥 ↔ 0 < +∞)) | |
10 | 8, 9 | mpbiri 259 | . . . . . 6 ⊢ (𝑥 = +∞ → 0 < 𝑥) |
11 | 5, 7, 10 | rspcedvd 3623 | . . . . 5 ⊢ (𝑥 = +∞ → ∃𝑦 ∈ ℝ 𝑦 < 𝑥) |
12 | 11 | a1d 25 | . . . 4 ⊢ (𝑥 = +∞ → (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) |
13 | breq2 5061 | . . . . 5 ⊢ (𝑥 = -∞ → (-∞ < 𝑥 ↔ -∞ < -∞)) | |
14 | mnfxr 10686 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
15 | nltmnf 12512 | . . . . . . 7 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
16 | 15 | pm2.21d 121 | . . . . . 6 ⊢ (-∞ ∈ ℝ* → (-∞ < -∞ → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) |
17 | 14, 16 | ax-mp 5 | . . . . 5 ⊢ (-∞ < -∞ → ∃𝑦 ∈ ℝ 𝑦 < 𝑥) |
18 | 13, 17 | syl6bi 254 | . . . 4 ⊢ (𝑥 = -∞ → (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) |
19 | 4, 12, 18 | 3jaoi 1419 | . . 3 ⊢ ((𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞) → (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) |
20 | 1, 19 | sylbi 218 | . 2 ⊢ (𝑥 ∈ ℝ* → (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) |
21 | 20 | rgen 3145 | 1 ⊢ ∀𝑥 ∈ ℝ* (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∨ w3o 1078 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 class class class wbr 5057 ℝcr 10524 0cc0 10525 +∞cpnf 10660 -∞cmnf 10661 ℝ*cxr 10662 < clt 10663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 |
This theorem is referenced by: infmremnf 12724 |
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