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Mirrors > Home > MPE Home > Th. List > Mathboxes > ipolub0 | Structured version Visualization version GIF version |
Description: The LUB of the empty set is the intersection of the base. (Contributed by Zhi Wang, 30-Sep-2024.) |
Ref | Expression |
---|---|
ipoglb0.i | ⊢ 𝐼 = (toInc‘𝐹) |
ipolub0.u | ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) |
ipolub0.f | ⊢ (𝜑 → ∩ 𝐹 ∈ 𝐹) |
ipolub0.v | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
Ref | Expression |
---|---|
ipolub0 | ⊢ (𝜑 → (𝑈‘∅) = ∩ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipoglb0.i | . 2 ⊢ 𝐼 = (toInc‘𝐹) | |
2 | ipolub0.v | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
3 | 0ss 4336 | . . 3 ⊢ ∅ ⊆ 𝐹 | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ⊆ 𝐹) |
5 | ipolub0.u | . 2 ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) | |
6 | uni0 4875 | . . . . . . . 8 ⊢ ∪ ∅ = ∅ | |
7 | 0ss 4336 | . . . . . . . 8 ⊢ ∅ ⊆ 𝑥 | |
8 | 6, 7 | eqsstri 3960 | . . . . . . 7 ⊢ ∪ ∅ ⊆ 𝑥 |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ 𝐹 → ∪ ∅ ⊆ 𝑥) |
10 | 9 | rabeqc 3624 | . . . . 5 ⊢ {𝑥 ∈ 𝐹 ∣ ∪ ∅ ⊆ 𝑥} = 𝐹 |
11 | 10 | eqcomi 2749 | . . . 4 ⊢ 𝐹 = {𝑥 ∈ 𝐹 ∣ ∪ ∅ ⊆ 𝑥} |
12 | 11 | inteqi 4889 | . . 3 ⊢ ∩ 𝐹 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ ∅ ⊆ 𝑥} |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → ∩ 𝐹 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ ∅ ⊆ 𝑥}) |
14 | ipolub0.f | . 2 ⊢ (𝜑 → ∩ 𝐹 ∈ 𝐹) | |
15 | 1, 2, 4, 5, 13, 14 | ipolub 46243 | 1 ⊢ (𝜑 → (𝑈‘∅) = ∩ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 {crab 3070 ⊆ wss 3892 ∅c0 4262 ∪ cuni 4845 ∩ cint 4885 ‘cfv 6432 lubclub 18025 toInccipo 18243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-fz 13239 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-tset 16979 df-ple 16980 df-ocomp 16981 df-proset 18011 df-poset 18029 df-lub 18062 df-ipo 18244 |
This theorem is referenced by: ipolub00 46248 |
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