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Mirrors > Home > MPE Home > Th. List > unfi | Structured version Visualization version GIF version |
Description: The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.) |
Ref | Expression |
---|---|
unfi | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diffi 8750 | . 2 ⊢ (𝐵 ∈ Fin → (𝐵 ∖ 𝐴) ∈ Fin) | |
2 | reeanv 3367 | . . . 4 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) ↔ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ∧ ∃𝑦 ∈ ω (𝐵 ∖ 𝐴) ≈ 𝑦)) | |
3 | isfi 8533 | . . . . 5 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
4 | isfi 8533 | . . . . 5 ⊢ ((𝐵 ∖ 𝐴) ∈ Fin ↔ ∃𝑦 ∈ ω (𝐵 ∖ 𝐴) ≈ 𝑦) | |
5 | 3, 4 | anbi12i 628 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin) ↔ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ∧ ∃𝑦 ∈ ω (𝐵 ∖ 𝐴) ≈ 𝑦)) |
6 | 2, 5 | bitr4i 280 | . . 3 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) ↔ (𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin)) |
7 | nnacl 8237 | . . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 +o 𝑦) ∈ ω) | |
8 | unfilem3 8784 | . . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) | |
9 | entr 8561 | . . . . . . . 8 ⊢ (((𝐵 ∖ 𝐴) ≈ 𝑦 ∧ 𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) | |
10 | 9 | expcom 416 | . . . . . . 7 ⊢ (𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥) → ((𝐵 ∖ 𝐴) ≈ 𝑦 → (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥))) |
11 | 8, 10 | syl 17 | . . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐵 ∖ 𝐴) ≈ 𝑦 → (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥))) |
12 | disjdif 4421 | . . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
13 | disjdif 4421 | . . . . . . . 8 ⊢ (𝑥 ∩ ((𝑥 +o 𝑦) ∖ 𝑥)) = ∅ | |
14 | unen 8596 | . . . . . . . 8 ⊢ (((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) ∧ ((𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ ∧ (𝑥 ∩ ((𝑥 +o 𝑦) ∖ 𝑥)) = ∅)) → (𝐴 ∪ (𝐵 ∖ 𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥))) | |
15 | 12, 13, 14 | mpanr12 703 | . . . . . . 7 ⊢ ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐴 ∪ (𝐵 ∖ 𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥))) |
16 | undif2 4425 | . . . . . . . . 9 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) | |
17 | 16 | a1i 11 | . . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)) |
18 | nnaword1 8255 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑥 ⊆ (𝑥 +o 𝑦)) | |
19 | undif 4430 | . . . . . . . . 9 ⊢ (𝑥 ⊆ (𝑥 +o 𝑦) ↔ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)) = (𝑥 +o 𝑦)) | |
20 | 18, 19 | sylib 220 | . . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)) = (𝑥 +o 𝑦)) |
21 | 17, 20 | breq12d 5079 | . . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ∪ (𝐵 ∖ 𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)) ↔ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦))) |
22 | 15, 21 | syl5ib 246 | . . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦))) |
23 | 11, 22 | sylan2d 606 | . . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦))) |
24 | breq2 5070 | . . . . . . 7 ⊢ (𝑧 = (𝑥 +o 𝑦) → ((𝐴 ∪ 𝐵) ≈ 𝑧 ↔ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦))) | |
25 | 24 | rspcev 3623 | . . . . . 6 ⊢ (((𝑥 +o 𝑦) ∈ ω ∧ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)) → ∃𝑧 ∈ ω (𝐴 ∪ 𝐵) ≈ 𝑧) |
26 | isfi 8533 | . . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin ↔ ∃𝑧 ∈ ω (𝐴 ∪ 𝐵) ≈ 𝑧) | |
27 | 25, 26 | sylibr 236 | . . . . 5 ⊢ (((𝑥 +o 𝑦) ∈ ω ∧ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)) → (𝐴 ∪ 𝐵) ∈ Fin) |
28 | 7, 23, 27 | syl6an 682 | . . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ∈ Fin)) |
29 | 28 | rexlimivv 3292 | . . 3 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ∈ Fin) |
30 | 6, 29 | sylbir 237 | . 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) |
31 | 1, 30 | sylan2 594 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 (class class class)co 7156 ωcom 7580 +o coa 8099 ≈ cen 8506 Fincfn 8509 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-oadd 8106 df-er 8289 df-en 8510 df-fin 8513 |
This theorem is referenced by: unfi2 8787 difinf 8788 xpfi 8789 prfi 8793 tpfi 8794 fnfi 8796 iunfi 8812 pwfilem 8818 fsuppun 8852 fsuppunfi 8853 ressuppfi 8859 fiin 8886 cantnfp1lem1 9141 ficardun2 9625 ackbij1lem6 9647 ackbij1lem16 9657 fin23lem28 9762 fin23lem30 9764 isfin1-3 9808 axcclem 9879 hashun 13744 hashunlei 13787 hashmap 13797 hashbclem 13811 hashf1lem1 13814 hashf1lem2 13815 hashf1 13816 fsumsplitsn 15100 fsummsnunz 15109 fsumsplitsnun 15110 incexclem 15191 isumltss 15203 fprodsplitsn 15343 lcmfunsnlem2lem1 15982 lcmfunsnlem2lem2 15983 lcmfunsnlem2 15984 lcmfun 15989 ramub1lem1 16362 fpwipodrs 17774 acsfiindd 17787 symgfisg 18596 gsumzunsnd 19076 gsumunsnfd 19077 psrbagaddcl 20150 mplsubg 20217 mpllss 20218 dsmmacl 20885 fctop 21612 uncmp 22011 bwth 22018 lfinun 22133 locfincmp 22134 comppfsc 22140 1stckgenlem 22161 ptbasin 22185 cfinfil 22501 fin1aufil 22540 alexsubALTlem3 22657 tmdgsum 22703 tsmsfbas 22736 tsmsgsum 22747 tsmsres 22752 tsmsxplem1 22761 prdsmet 22980 prdsbl 23101 icccmplem2 23431 rrxmval 24008 rrxmet 24011 rrxdstprj1 24012 ovolfiniun 24102 volfiniun 24148 fta1glem2 24760 fta1lem 24896 aannenlem2 24918 aalioulem2 24922 dchrfi 25831 usgrfilem 27109 ffsrn 30465 eulerpartlemt 31629 ballotlemgun 31782 hgt750lemb 31927 hgt750leme 31929 lindsenlbs 34902 poimirlem31 34938 poimirlem32 34939 itg2addnclem2 34959 ftc1anclem7 34988 ftc1anc 34990 prdsbnd 35086 pclfinN 37051 elrfi 39311 mzpcompact2lem 39368 eldioph2 39379 lsmfgcl 39694 fiuneneq 39817 unfid 41442 dvmptfprodlem 42249 dvnprodlem2 42252 fourierdlem50 42461 fourierdlem51 42462 fourierdlem54 42465 fourierdlem76 42487 fourierdlem80 42491 fourierdlem102 42513 fourierdlem103 42514 fourierdlem104 42515 fourierdlem114 42525 sge0resplit 42708 sge0iunmptlemfi 42715 sge0xaddlem1 42735 hoiprodp1 42890 sge0hsphoire 42891 hoidmvlelem1 42897 hoidmvlelem2 42898 hoidmvlelem5 42901 hspmbllem2 42929 fsummmodsnunz 43555 mndpsuppfi 44443 |
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