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Mirrors > Home > MPE Home > Th. List > fpwipodrs | Structured version Visualization version GIF version |
Description: The finite subsets of any set are directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
Ref | Expression |
---|---|
fpwipodrs | ⊢ (𝐴 ∈ 𝑉 → (toInc‘(𝒫 𝐴 ∩ Fin)) ∈ Dirset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 5080 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
2 | inex1g 5028 | . . 3 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ∩ Fin) ∈ V) |
4 | 0elpw 5058 | . . . 4 ⊢ ∅ ∈ 𝒫 𝐴 | |
5 | 0fin 8463 | . . . 4 ⊢ ∅ ∈ Fin | |
6 | elin 4025 | . . . 4 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ∈ Fin)) | |
7 | 4, 5, 6 | mpbir2an 702 | . . 3 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin) |
8 | ne0i 4152 | . . 3 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) → (𝒫 𝐴 ∩ Fin) ≠ ∅) | |
9 | 7, 8 | mp1i 13 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ∩ Fin) ≠ ∅) |
10 | elin 4025 | . . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin)) | |
11 | elin 4025 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ Fin)) | |
12 | elpwi 4390 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
13 | elpwi 4390 | . . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴) | |
14 | 12, 13 | anim12i 606 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴)) |
15 | unss 4016 | . . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ↔ (𝑥 ∪ 𝑦) ⊆ 𝐴) | |
16 | vex 3417 | . . . . . . . . . . . 12 ⊢ 𝑥 ∈ V | |
17 | vex 3417 | . . . . . . . . . . . 12 ⊢ 𝑦 ∈ V | |
18 | 16, 17 | unex 7221 | . . . . . . . . . . 11 ⊢ (𝑥 ∪ 𝑦) ∈ V |
19 | 18 | elpw 4386 | . . . . . . . . . 10 ⊢ ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐴 ↔ (𝑥 ∪ 𝑦) ⊆ 𝐴) |
20 | 15, 19 | bitr4i 270 | . . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ↔ (𝑥 ∪ 𝑦) ∈ 𝒫 𝐴) |
21 | 14, 20 | sylib 210 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (𝑥 ∪ 𝑦) ∈ 𝒫 𝐴) |
22 | 21 | ad2ant2r 753 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ Fin)) → (𝑥 ∪ 𝑦) ∈ 𝒫 𝐴) |
23 | unfi 8502 | . . . . . . . 8 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin) | |
24 | 23 | ad2ant2l 752 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ Fin)) → (𝑥 ∪ 𝑦) ∈ Fin) |
25 | 22, 24 | elind 4027 | . . . . . 6 ⊢ (((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ Fin)) → (𝑥 ∪ 𝑦) ∈ (𝒫 𝐴 ∩ Fin)) |
26 | 10, 11, 25 | syl2anb 591 | . . . . 5 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑥 ∪ 𝑦) ∈ (𝒫 𝐴 ∩ Fin)) |
27 | ssid 3848 | . . . . 5 ⊢ (𝑥 ∪ 𝑦) ⊆ (𝑥 ∪ 𝑦) | |
28 | sseq2 3852 | . . . . . 6 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → ((𝑥 ∪ 𝑦) ⊆ 𝑧 ↔ (𝑥 ∪ 𝑦) ⊆ (𝑥 ∪ 𝑦))) | |
29 | 28 | rspcev 3526 | . . . . 5 ⊢ (((𝑥 ∪ 𝑦) ∈ (𝒫 𝐴 ∩ Fin) ∧ (𝑥 ∪ 𝑦) ⊆ (𝑥 ∪ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑥 ∪ 𝑦) ⊆ 𝑧) |
30 | 26, 27, 29 | sylancl 580 | . . . 4 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑥 ∪ 𝑦) ⊆ 𝑧) |
31 | 30 | rgen2a 3186 | . . 3 ⊢ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)∀𝑦 ∈ (𝒫 𝐴 ∩ Fin)∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑥 ∪ 𝑦) ⊆ 𝑧 |
32 | 31 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)∀𝑦 ∈ (𝒫 𝐴 ∩ Fin)∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑥 ∪ 𝑦) ⊆ 𝑧) |
33 | isipodrs 17521 | . 2 ⊢ ((toInc‘(𝒫 𝐴 ∩ Fin)) ∈ Dirset ↔ ((𝒫 𝐴 ∩ Fin) ∈ V ∧ (𝒫 𝐴 ∩ Fin) ≠ ∅ ∧ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)∀𝑦 ∈ (𝒫 𝐴 ∩ Fin)∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑥 ∪ 𝑦) ⊆ 𝑧)) | |
34 | 3, 9, 32, 33 | syl3anbrc 1447 | 1 ⊢ (𝐴 ∈ 𝑉 → (toInc‘(𝒫 𝐴 ∩ Fin)) ∈ Dirset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 Vcvv 3414 ∪ cun 3796 ∩ cin 3797 ⊆ wss 3798 ∅c0 4146 𝒫 cpw 4380 ‘cfv 6127 Fincfn 8228 Dirsetcdrs 17287 toInccipo 17511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-tset 16331 df-ple 16332 df-ocomp 16333 df-proset 17288 df-drs 17289 df-poset 17306 df-ipo 17512 |
This theorem is referenced by: isacs5lem 17529 isnacs3 38112 |
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