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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 13158 | . . 3 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) | |
| 2 | reltre 13382 | . . . . . 6 ⊢ ∀𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑦 < 𝑥 | |
| 3 | 2 | rspec 3250 | . . . . 5 ⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ 𝑦 < 𝑥) | 
| 4 | 3 | a1d 25 | . . . 4 ⊢ (𝑥 ∈ ℝ → (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) | 
| 5 | breq1 5146 | . . . . . 6 ⊢ (𝑦 = 0 → (𝑦 < 𝑥 ↔ 0 < 𝑥)) | |
| 6 | 0red 11264 | . . . . . 6 ⊢ (𝑥 = +∞ → 0 ∈ ℝ) | |
| 7 | 0ltpnf 13164 | . . . . . . 7 ⊢ 0 < +∞ | |
| 8 | breq2 5147 | . . . . . . 7 ⊢ (𝑥 = +∞ → (0 < 𝑥 ↔ 0 < +∞)) | |
| 9 | 7, 8 | mpbiri 258 | . . . . . 6 ⊢ (𝑥 = +∞ → 0 < 𝑥) | 
| 10 | 5, 6, 9 | rspcedvdw 3625 | . . . . 5 ⊢ (𝑥 = +∞ → ∃𝑦 ∈ ℝ 𝑦 < 𝑥) | 
| 11 | 10 | a1d 25 | . . . 4 ⊢ (𝑥 = +∞ → (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) | 
| 12 | breq2 5147 | . . . . 5 ⊢ (𝑥 = -∞ → (-∞ < 𝑥 ↔ -∞ < -∞)) | |
| 13 | mnfxr 11318 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
| 14 | nltmnf 13171 | . . . . . . 7 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
| 15 | 14 | pm2.21d 121 | . . . . . 6 ⊢ (-∞ ∈ ℝ* → (-∞ < -∞ → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) | 
| 16 | 13, 15 | ax-mp 5 | . . . . 5 ⊢ (-∞ < -∞ → ∃𝑦 ∈ ℝ 𝑦 < 𝑥) | 
| 17 | 12, 16 | biimtrdi 253 | . . . 4 ⊢ (𝑥 = -∞ → (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) | 
| 18 | 4, 11, 17 | 3jaoi 1430 | . . 3 ⊢ ((𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞) → (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) | 
| 19 | 1, 18 | sylbi 217 | . 2 ⊢ (𝑥 ∈ ℝ* → (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)) | 
| 20 | 19 | rgen 3063 | 1 ⊢ ∀𝑥 ∈ ℝ* (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 class class class wbr 5143 ℝcr 11154 0cc0 11155 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 < clt 11295 | 
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 | 
| This theorem is referenced by: infmremnf 13385 | 
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