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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 5281 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
2 | inex1g 5225 | . . 3 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ∩ Fin) ∈ V) |
4 | 0elpw 5258 | . . . 4 ⊢ ∅ ∈ 𝒫 𝐴 | |
5 | 0fin 8748 | . . . 4 ⊢ ∅ ∈ Fin | |
6 | 4, 5 | elini 4172 | . . 3 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin) |
7 | ne0i 4302 | . . 3 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) → (𝒫 𝐴 ∩ Fin) ≠ ∅) | |
8 | 6, 7 | mp1i 13 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ∩ Fin) ≠ ∅) |
9 | elin 4171 | . . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin)) | |
10 | elin 4171 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ Fin)) | |
11 | pwuncl 7494 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (𝑥 ∪ 𝑦) ∈ 𝒫 𝐴) | |
12 | 11 | ad2ant2r 745 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ Fin)) → (𝑥 ∪ 𝑦) ∈ 𝒫 𝐴) |
13 | unfi 8787 | . . . . . . . 8 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin) | |
14 | 13 | ad2ant2l 744 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ Fin)) → (𝑥 ∪ 𝑦) ∈ Fin) |
15 | 12, 14 | elind 4173 | . . . . . 6 ⊢ (((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ Fin)) → (𝑥 ∪ 𝑦) ∈ (𝒫 𝐴 ∩ Fin)) |
16 | 9, 10, 15 | syl2anb 599 | . . . . 5 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑥 ∪ 𝑦) ∈ (𝒫 𝐴 ∩ Fin)) |
17 | ssid 3991 | . . . . 5 ⊢ (𝑥 ∪ 𝑦) ⊆ (𝑥 ∪ 𝑦) | |
18 | sseq2 3995 | . . . . . 6 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → ((𝑥 ∪ 𝑦) ⊆ 𝑧 ↔ (𝑥 ∪ 𝑦) ⊆ (𝑥 ∪ 𝑦))) | |
19 | 18 | rspcev 3625 | . . . . 5 ⊢ (((𝑥 ∪ 𝑦) ∈ (𝒫 𝐴 ∩ Fin) ∧ (𝑥 ∪ 𝑦) ⊆ (𝑥 ∪ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑥 ∪ 𝑦) ⊆ 𝑧) |
20 | 16, 17, 19 | sylancl 588 | . . . 4 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑥 ∪ 𝑦) ⊆ 𝑧) |
21 | 20 | rgen2 3205 | . . 3 ⊢ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)∀𝑦 ∈ (𝒫 𝐴 ∩ Fin)∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑥 ∪ 𝑦) ⊆ 𝑧 |
22 | 21 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)∀𝑦 ∈ (𝒫 𝐴 ∩ Fin)∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑥 ∪ 𝑦) ⊆ 𝑧) |
23 | isipodrs 17773 | . 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)) | |
24 | 3, 8, 22, 23 | syl3anbrc 1339 | 1 ⊢ (𝐴 ∈ 𝑉 → (toInc‘(𝒫 𝐴 ∩ Fin)) ∈ Dirset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ∪ cun 3936 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 ‘cfv 6357 Fincfn 8511 Dirsetcdrs 17539 toInccipo 17763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-tset 16586 df-ple 16587 df-ocomp 16588 df-proset 17540 df-drs 17541 df-poset 17558 df-ipo 17764 |
This theorem is referenced by: isacs5lem 17781 isnacs3 39314 |
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