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| Mirrors > Home > MPE Home > Th. List > xrsupexmnf | Structured version Visualization version GIF version | ||
| Description: Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.) | 
| Ref | Expression | 
|---|---|
| xrsupexmnf | ⊢ (∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elun 4153 | . . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∪ {-∞}) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ {-∞})) | |
| 2 | simpr 484 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) | |
| 3 | velsn 4642 | . . . . . . . . 9 ⊢ (𝑦 ∈ {-∞} ↔ 𝑦 = -∞) | |
| 4 | nltmnf 13171 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞) | |
| 5 | breq2 5147 | . . . . . . . . . . 11 ⊢ (𝑦 = -∞ → (𝑥 < 𝑦 ↔ 𝑥 < -∞)) | |
| 6 | 5 | notbid 318 | . . . . . . . . . 10 ⊢ (𝑦 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < -∞)) | 
| 7 | 4, 6 | syl5ibrcom 247 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ* → (𝑦 = -∞ → ¬ 𝑥 < 𝑦)) | 
| 8 | 3, 7 | biimtrid 242 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦)) | 
| 9 | 8 | adantr 480 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ {-∞} → ¬ 𝑥 < 𝑦)) | 
| 10 | 2, 9 | jaod 860 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ {-∞}) → ¬ 𝑥 < 𝑦)) | 
| 11 | 1, 10 | biimtrid 242 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦)) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦)) | 
| 12 | 11 | ex 412 | . . . 4 ⊢ (𝑥 ∈ ℝ* → ((𝑦 ∈ 𝐴 → ¬ 𝑥 < 𝑦) → (𝑦 ∈ (𝐴 ∪ {-∞}) → ¬ 𝑥 < 𝑦))) | 
| 13 | 12 | ralimdv2 3163 | . . 3 ⊢ (𝑥 ∈ ℝ* → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 → ∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦)) | 
| 14 | elun1 4182 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (𝐴 ∪ {-∞})) | |
| 15 | 14 | anim1i 615 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 < 𝑧) → (𝑧 ∈ (𝐴 ∪ {-∞}) ∧ 𝑦 < 𝑧)) | 
| 16 | 15 | reximi2 3079 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧) | 
| 17 | 16 | imim2i 16 | . . . 4 ⊢ ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)) | 
| 18 | 17 | ralimi 3083 | . . 3 ⊢ (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)) | 
| 19 | 13, 18 | anim12d1 610 | . 2 ⊢ (𝑥 ∈ ℝ* → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → (∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)))) | 
| 20 | 19 | reximia 3081 | 1 ⊢ (∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∪ cun 3949 {csn 4626 class class class wbr 5143 -∞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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-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-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-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 | 
| This theorem is referenced by: xrsupss 13351 | 
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