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Theorem unfi 8227

Description: The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.)

**Assertion**
Ref	Expression
unfi	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Proof of Theorem unfi Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		diffi 8192	. 2 ⊢ (𝐵 ∈ Fin → (𝐵 ∖ 𝐴) ∈ Fin)
2		reeanv 3107	. . . 4 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) ↔ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ∧ ∃𝑦 ∈ ω (𝐵 ∖ 𝐴) ≈ 𝑦))
3		isfi 7979	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		isfi 7979	. . . . 5 ⊢ ((𝐵 ∖ 𝐴) ∈ Fin ↔ ∃𝑦 ∈ ω (𝐵 ∖ 𝐴) ≈ 𝑦)
5	3, 4	anbi12i 733	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin) ↔ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ∧ ∃𝑦 ∈ ω (𝐵 ∖ 𝐴) ≈ 𝑦))
6	2, 5	bitr4i 267	. . 3 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) ↔ (𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin))
7		nnacl 7691	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 +_𝑜 𝑦) ∈ ω)
8		unfilem3 8226	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑦 ≈ ((𝑥 +_𝑜 𝑦) ∖ 𝑥))
9		entr 8008	. . . . . . . 8 ⊢ (((𝐵 ∖ 𝐴) ≈ 𝑦 ∧ 𝑦 ≈ ((𝑥 +_𝑜 𝑦) ∖ 𝑥)) → (𝐵 ∖ 𝐴) ≈ ((𝑥 +_𝑜 𝑦) ∖ 𝑥))
10	9	expcom 451	. . . . . . 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 4040	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
13		disjdif 4040	. . . . . . . 8 ⊢ (𝑥 ∩ ((𝑥 +_𝑜 𝑦) ∖ 𝑥)) = ∅
14		unen 8040	. . . . . . . 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 721	. . . . . . 7 ⊢ ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ ((𝑥 +_𝑜 𝑦) ∖ 𝑥)) → (𝐴 ∪ (𝐵 ∖ 𝐴)) ≈ (𝑥 ∪ ((𝑥 +_𝑜 𝑦) ∖ 𝑥)))
16		undif2 4044	. . . . . . . . 9 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
17	16	a1i 11	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵))
18		nnaword1 7709	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑥 ⊆ (𝑥 +_𝑜 𝑦))
19		undif 4049	. . . . . . . . 9 ⊢ (𝑥 ⊆ (𝑥 +_𝑜 𝑦) ↔ (𝑥 ∪ ((𝑥 +_𝑜 𝑦) ∖ 𝑥)) = (𝑥 +_𝑜 𝑦))
20	18, 19	sylib 208	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ∪ ((𝑥 +_𝑜 𝑦) ∖ 𝑥)) = (𝑥 +_𝑜 𝑦))
21	17, 20	breq12d 4666	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ∪ (𝐵 ∖ 𝐴)) ≈ (𝑥 ∪ ((𝑥 +_𝑜 𝑦) ∖ 𝑥)) ↔ (𝐴 ∪ 𝐵) ≈ (𝑥 +_𝑜 𝑦)))
22	15, 21	syl5ib 234	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ ((𝑥 +_𝑜 𝑦) ∖ 𝑥)) → (𝐴 ∪ 𝐵) ≈ (𝑥 +_𝑜 𝑦)))
23	11, 22	sylan2d 499	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ≈ (𝑥 +_𝑜 𝑦)))
24		breq2 4657	. . . . . . 7 ⊢ (𝑧 = (𝑥 +_𝑜 𝑦) → ((𝐴 ∪ 𝐵) ≈ 𝑧 ↔ (𝐴 ∪ 𝐵) ≈ (𝑥 +_𝑜 𝑦)))
25	24	rspcev 3309	. . . . . 6 ⊢ (((𝑥 +_𝑜 𝑦) ∈ ω ∧ (𝐴 ∪ 𝐵) ≈ (𝑥 +_𝑜 𝑦)) → ∃𝑧 ∈ ω (𝐴 ∪ 𝐵) ≈ 𝑧)
26		isfi 7979	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin ↔ ∃𝑧 ∈ ω (𝐴 ∪ 𝐵) ≈ 𝑧)
27	25, 26	sylibr 224	. . . . 5 ⊢ (((𝑥 +_𝑜 𝑦) ∈ ω ∧ (𝐴 ∪ 𝐵) ≈ (𝑥 +_𝑜 𝑦)) → (𝐴 ∪ 𝐵) ∈ Fin)
28	7, 23, 27	syl6an 568	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ∈ Fin))
29	28	rexlimivv 3036	. . 3 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ∈ Fin)
30	6, 29	sylbir 225	. 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
31	1, 30	sylan2 491	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 (class class class)co 6650 ωcom 7065 +_𝑜 coa 7557 ≈ cen 7952 Fincfn 7955

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-er 7742 df-en 7956 df-fin 7959

This theorem is referenced by: unfi2 8229 difinf 8230 xpfi 8231 prfi 8235 tpfi 8236 fnfi 8238 iunfi 8254 pwfilem 8260 fsuppun 8294 fsuppunfi 8295 ressuppfi 8301 fiin 8328 cantnfp1lem1 8575 ficardun2 9025 ackbij1lem6 9047 ackbij1lem16 9057 fin23lem28 9162 fin23lem30 9164 isfin1-3 9208 axcclem 9279 hashun 13171 hashunlei 13212 hashmap 13222 hashbclem 13236 hashf1lem1 13239 hashf1lem2 13240 hashf1 13241 fsumsplitsn 14474 fsummsnunz 14483 fsumsplitsnun 14484 fsummsnunzOLD 14485 fsumsplitsnunOLD 14486 incexclem 14568 isumltss 14580 fprodsplitsn 14720 lcmfunsnlem2lem1 15351 lcmfunsnlem2lem2 15352 lcmfunsnlem2 15353 lcmfun 15358 ramub1lem1 15730 fpwipodrs 17164 acsfiindd 17177 symgfisg 17888 gsumzunsnd 18355 gsumunsnfd 18356 psrbagaddcl 19370 mplsubg 19437 mpllss 19438 dsmmacl 20085 fctop 20808 uncmp 21206 bwth 21213 lfinun 21328 locfincmp 21329 comppfsc 21335 1stckgenlem 21356 ptbasin 21380 cfinfil 21697 fin1aufil 21736 alexsubALTlem3 21853 tmdgsum 21899 tsmsfbas 21931 tsmsgsum 21942 tsmsres 21947 tsmsxplem1 21956 prdsmet 22175 prdsbl 22296 icccmplem2 22626 rrxmval 23188 rrxmet 23191 rrxdstprj1 23192 ovolfiniun 23269 volfiniun 23315 fta1glem2 23926 fta1lem 24062 aannenlem2 24084 aalioulem2 24088 dchrfi 24980 usgrfilem 26219 ffsrn 29504 eulerpartlemt 30433 ballotlemgun 30586 hgt750lemb 30734 hgt750leme 30736 lindsenlbs 33404 poimirlem31 33440 poimirlem32 33441 itg2addnclem2 33462 ftc1anclem7 33491 ftc1anc 33493 prdsbnd 33592 pclfinN 35186 elrfi 37257 mzpcompact2lem 37314 eldioph2 37325 lsmfgcl 37644 fiuneneq 37775 unfid 39345 dvmptfprodlem 40159 dvnprodlem2 40162 fourierdlem50 40373 fourierdlem51 40374 fourierdlem54 40377 fourierdlem76 40399 fourierdlem80 40403 fourierdlem102 40425 fourierdlem103 40426 fourierdlem104 40427 fourierdlem114 40437 sge0resplit 40623 sge0iunmptlemfi 40630 sge0xaddlem1 40650 hoiprodp1 40802 sge0hsphoire 40803 hoidmvlelem1 40809 hoidmvlelem2 40810 hoidmvlelem5 40813 hspmbllem2 40841 fsummmodsnunz 41345 mndpsuppfi 42156

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