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Mirrors > Home > MPE Home > Th. List > rpltrp | Structured version Visualization version GIF version |
Description: For all positive real numbers there is a smaller positive real number. (Contributed by AV, 5-Sep-2020.) |
Ref | Expression |
---|---|
rpltrp | ⊢ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ+ 𝑦 < 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rphalfcl 12997 | . . 3 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+) | |
2 | breq1 5141 | . . . 4 ⊢ (𝑦 = (𝑥 / 2) → (𝑦 < 𝑥 ↔ (𝑥 / 2) < 𝑥)) | |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑦 = (𝑥 / 2)) → (𝑦 < 𝑥 ↔ (𝑥 / 2) < 𝑥)) |
4 | rphalflt 12999 | . . 3 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) < 𝑥) | |
5 | 1, 3, 4 | rspcedvd 3606 | . 2 ⊢ (𝑥 ∈ ℝ+ → ∃𝑦 ∈ ℝ+ 𝑦 < 𝑥) |
6 | 5 | rgen 3055 | 1 ⊢ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ+ 𝑦 < 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 class class class wbr 5138 (class class class)co 7401 < clt 11244 / cdiv 11867 2c2 12263 ℝ+crp 12970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-2 12271 df-rp 12971 |
This theorem is referenced by: infmrp1 13319 |
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