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Mirrors > Home > MPE Home > Th. List > suprcld | Structured version Visualization version GIF version |
Description: Natural deduction form of suprcl 12073. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
suprcld.2 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
suprcld.1 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
suprcld.4 | ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
Ref | Expression |
---|---|
suprcld | ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suprcld.2 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | suprcld.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
3 | suprcld.4 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) | |
4 | suprcl 12073 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ) | |
5 | 1, 2, 3, 4 | syl3anc 1371 | 1 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ⊆ wss 3908 ∅c0 4280 class class class wbr 5103 supcsup 9334 ℝcr 11008 < clt 11147 ≤ cle 11148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 |
This theorem is referenced by: supaddc 12080 supadd 12081 flval3 13674 supcvg 15695 ruclem12 16077 prmreclem6 16747 icccmplem2 24132 icccmplem3 24133 reconnlem2 24136 ivthlem2 24762 ivthlem3 24763 ioombl1lem4 24871 mbfsup 24974 mbflimsup 24976 itg2monolem1 25061 itg2mono 25064 itg2cnlem1 25072 c1liplem1 25306 imo72b2lem0 42343 imo72b2 42350 suprclrnmpt 43378 |
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