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Mirrors > Home > MPE Home > Th. List > Mathboxes > ipolub00 | Structured version Visualization version GIF version |
Description: The LUB of the empty set is the empty set if it is contained. (Contributed by Zhi Wang, 30-Sep-2024.) |
Ref | Expression |
---|---|
ipoglb0.i | ⊢ 𝐼 = (toInc‘𝐹) |
ipolub00.u | ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) |
ipolub00.f | ⊢ (𝜑 → ∅ ∈ 𝐹) |
Ref | Expression |
---|---|
ipolub00 | ⊢ (𝜑 → (𝑈‘∅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipoglb0.i | . . . 4 ⊢ 𝐼 = (toInc‘𝐹) | |
2 | ipolub00.u | . . . . 5 ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) | |
3 | 2 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝑈 = (lub‘𝐼)) |
4 | ipolub00.f | . . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝐹) | |
5 | int0el 4876 | . . . . . . 7 ⊢ (∅ ∈ 𝐹 → ∩ 𝐹 = ∅) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∩ 𝐹 = ∅) |
7 | 6, 4 | eqeltrd 2831 | . . . . 5 ⊢ (𝜑 → ∩ 𝐹 ∈ 𝐹) |
8 | 7 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → ∩ 𝐹 ∈ 𝐹) |
9 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐹 ∈ V) | |
10 | 1, 3, 8, 9 | ipolub0 45894 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝑈‘∅) = ∩ 𝐹) |
11 | 6 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → ∩ 𝐹 = ∅) |
12 | 10, 11 | eqtrd 2771 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝑈‘∅) = ∅) |
13 | 2 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → 𝑈 = (lub‘𝐼)) |
14 | fvprc 6687 | . . . . . . . 8 ⊢ (¬ 𝐹 ∈ V → (toInc‘𝐹) = ∅) | |
15 | 14 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (toInc‘𝐹) = ∅) |
16 | 1, 15 | syl5eq 2783 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → 𝐼 = ∅) |
17 | 16 | fveq2d 6699 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (lub‘𝐼) = (lub‘∅)) |
18 | 13, 17 | eqtrd 2771 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → 𝑈 = (lub‘∅)) |
19 | 18 | fveq1d 6697 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝑈‘∅) = ((lub‘∅)‘∅)) |
20 | rex0 4258 | . . . . . 6 ⊢ ¬ ∃𝑥 ∈ ∅ (∀𝑦 ∈ ∅ 𝑦(le‘∅)𝑥 ∧ ∀𝑧 ∈ ∅ (∀𝑦 ∈ ∅ 𝑦(le‘∅)𝑧 → 𝑥(le‘∅)𝑧)) | |
21 | 20 | intnan 490 | . . . . 5 ⊢ ¬ (∅ ⊆ ∅ ∧ ∃𝑥 ∈ ∅ (∀𝑦 ∈ ∅ 𝑦(le‘∅)𝑥 ∧ ∀𝑧 ∈ ∅ (∀𝑦 ∈ ∅ 𝑦(le‘∅)𝑧 → 𝑥(le‘∅)𝑧))) |
22 | base0 16726 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
23 | eqid 2736 | . . . . . 6 ⊢ (le‘∅) = (le‘∅) | |
24 | eqid 2736 | . . . . . 6 ⊢ (lub‘∅) = (lub‘∅) | |
25 | biid 264 | . . . . . 6 ⊢ ((∀𝑦 ∈ ∅ 𝑦(le‘∅)𝑥 ∧ ∀𝑧 ∈ ∅ (∀𝑦 ∈ ∅ 𝑦(le‘∅)𝑧 → 𝑥(le‘∅)𝑧)) ↔ (∀𝑦 ∈ ∅ 𝑦(le‘∅)𝑥 ∧ ∀𝑧 ∈ ∅ (∀𝑦 ∈ ∅ 𝑦(le‘∅)𝑧 → 𝑥(le‘∅)𝑧))) | |
26 | 0pos 17782 | . . . . . . 7 ⊢ ∅ ∈ Poset | |
27 | 26 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → ∅ ∈ Poset) |
28 | 22, 23, 24, 25, 27 | lubeldm2 45866 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (∅ ∈ dom (lub‘∅) ↔ (∅ ⊆ ∅ ∧ ∃𝑥 ∈ ∅ (∀𝑦 ∈ ∅ 𝑦(le‘∅)𝑥 ∧ ∀𝑧 ∈ ∅ (∀𝑦 ∈ ∅ 𝑦(le‘∅)𝑧 → 𝑥(le‘∅)𝑧))))) |
29 | 21, 28 | mtbiri 330 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → ¬ ∅ ∈ dom (lub‘∅)) |
30 | ndmfv 6725 | . . . 4 ⊢ (¬ ∅ ∈ dom (lub‘∅) → ((lub‘∅)‘∅) = ∅) | |
31 | 29, 30 | syl 17 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → ((lub‘∅)‘∅) = ∅) |
32 | 19, 31 | eqtrd 2771 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝑈‘∅) = ∅) |
33 | 12, 32 | pm2.61dan 813 | 1 ⊢ (𝜑 → (𝑈‘∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 Vcvv 3398 ⊆ wss 3853 ∅c0 4223 ∩ cint 4845 class class class wbr 5039 dom cdm 5536 ‘cfv 6358 lecple 16756 Posetcpo 17768 lubclub 17770 toInccipo 17987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-tset 16768 df-ple 16769 df-ocomp 16770 df-proset 17756 df-poset 17774 df-lub 17806 df-ipo 17988 |
This theorem is referenced by: (None) |
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