Theorem unfi 8741

Description: The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.) Avoid ax-pow 5234. (Revised by BTernaryTau, 7-Aug-2024.)

Assertion

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

Proof of Theorem unfi

Dummy variables 𝑥 𝑦 𝑧 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uneq2 4062	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ∪ 𝑥) = (𝐴 ∪ ∅))
2	1	eleq1d 2836	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ∪ 𝑥) ∈ Fin ↔ (𝐴 ∪ ∅) ∈ Fin))
3	2	imbi2d 344	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ∈ Fin → (𝐴 ∪ 𝑥) ∈ Fin) ↔ (𝐴 ∈ Fin → (𝐴 ∪ ∅) ∈ Fin)))
4		uneq2 4062	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∪ 𝑥) = (𝐴 ∪ 𝑦))
5	4	eleq1d 2836	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ∪ 𝑥) ∈ Fin ↔ (𝐴 ∪ 𝑦) ∈ Fin))
6	5	imbi2d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ Fin → (𝐴 ∪ 𝑥) ∈ Fin) ↔ (𝐴 ∈ Fin → (𝐴 ∪ 𝑦) ∈ Fin)))
7		uneq2 4062	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ∪ 𝑥) = (𝐴 ∪ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2836	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∪ 𝑥) ∈ Fin ↔ (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
9	8	imbi2d 344	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∈ Fin → (𝐴 ∪ 𝑥) ∈ Fin) ↔ (𝐴 ∈ Fin → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)))
10		uneq2 4062	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∪ 𝑥) = (𝐴 ∪ 𝐵))
11	10	eleq1d 2836	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∪ 𝑥) ∈ Fin ↔ (𝐴 ∪ 𝐵) ∈ Fin))
12	11	imbi2d 344	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ Fin → (𝐴 ∪ 𝑥) ∈ Fin) ↔ (𝐴 ∈ Fin → (𝐴 ∪ 𝐵) ∈ Fin)))
13		un0 4286	. . . . 5 ⊢ (𝐴 ∪ ∅) = 𝐴
14	13	eleq1i 2842	. . . 4 ⊢ ((𝐴 ∪ ∅) ∈ Fin ↔ 𝐴 ∈ Fin)
15	14	biimpri 231	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ∪ ∅) ∈ Fin)
16		snssi 4698	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 → {𝑧} ⊆ 𝐴)
17		ssequn2 4088	. . . . . . . . . . . . . 14 ⊢ ({𝑧} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑧}) = 𝐴)
18	17	biimpi 219	. . . . . . . . . . . . 13 ⊢ ({𝑧} ⊆ 𝐴 → (𝐴 ∪ {𝑧}) = 𝐴)
19	18	uneq2d 4068	. . . . . . . . . . . 12 ⊢ ({𝑧} ⊆ 𝐴 → (𝑦 ∪ (𝐴 ∪ {𝑧})) = (𝑦 ∪ 𝐴))
20		un12 4072	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ (𝑦 ∪ {𝑧})) = (𝑦 ∪ (𝐴 ∪ {𝑧}))
21		uncom 4058	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ 𝑦) = (𝑦 ∪ 𝐴)
22	19, 20, 21	3eqtr4g 2818	. . . . . . . . . . 11 ⊢ ({𝑧} ⊆ 𝐴 → (𝐴 ∪ (𝑦 ∪ {𝑧})) = (𝐴 ∪ 𝑦))
23	16, 22	syl 17	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → (𝐴 ∪ (𝑦 ∪ {𝑧})) = (𝐴 ∪ 𝑦))
24	23	eleq1d 2836	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → ((𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin ↔ (𝐴 ∪ 𝑦) ∈ Fin))
25	24	biimprd 251	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → ((𝐴 ∪ 𝑦) ∈ Fin → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
26	25	adantld 494	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((¬ 𝑧 ∈ 𝑦 ∧ (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
27		isfi 8551	. . . . . . . . . . 11 ⊢ ((𝐴 ∪ 𝑦) ∈ Fin ↔ ∃𝑤 ∈ ω (𝐴 ∪ 𝑦) ≈ 𝑤)
28	27	biimpi 219	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝑦) ∈ Fin → ∃𝑤 ∈ ω (𝐴 ∪ 𝑦) ≈ 𝑤)
29		r19.41v 3265	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ ω ((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)) ↔ (∃𝑤 ∈ ω (𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)))
30		disjsn 4604	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∪ 𝑦) ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ (𝐴 ∪ 𝑦))
31		elun 4054	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝐴 ∪ 𝑦) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
32	31	notbii 323	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑧 ∈ (𝐴 ∪ 𝑦) ↔ ¬ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
33		pm4.56 986	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦) ↔ ¬ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
34	32, 33	bitr4i 281	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑧 ∈ (𝐴 ∪ 𝑦) ↔ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦))
35	30, 34	sylbbr 239	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐴 ∪ 𝑦) ∩ {𝑧}) = ∅)
36		nnord 7587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ω → Ord 𝑤)
37		orddisj 6207	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑤 → (𝑤 ∩ {𝑤}) = ∅)
38	36, 37	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ω → (𝑤 ∩ {𝑤}) = ∅)
39		en2sn 8612	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → {𝑧} ≈ {𝑤})
40	39	el2v 3417	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑧} ≈ {𝑤}
41		unen 8616	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ {𝑧} ≈ {𝑤}) ∧ (((𝐴 ∪ 𝑦) ∩ {𝑧}) = ∅ ∧ (𝑤 ∩ {𝑤}) = ∅)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
42	40, 41	mpanl2 700	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (((𝐴 ∪ 𝑦) ∩ {𝑧}) = ∅ ∧ (𝑤 ∩ {𝑤}) = ∅)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
43	38, 42	sylanr2 682	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (((𝐴 ∪ 𝑦) ∩ {𝑧}) = ∅ ∧ 𝑤 ∈ ω)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
44	35, 43	sylanr1 681	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ ((¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑤 ∈ ω)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
45	44	3impb 1112	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑤 ∈ ω) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
46	45	3comr 1122	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ω ∧ (𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
47	46	3expb 1117	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ ((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦))) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
48		unass 4071	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∪ 𝑦) ∪ {𝑧}) = (𝐴 ∪ (𝑦 ∪ {𝑧}))
49		df-suc 6175	. . . . . . . . . . . . . . . . 17 ⊢ suc 𝑤 = (𝑤 ∪ {𝑤})
50		peano2 7601	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ω → suc 𝑤 ∈ ω)
51	49, 50	eqeltrrid 2857	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ω → (𝑤 ∪ {𝑤}) ∈ ω)
52		breq2 5036	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑤 ∪ {𝑤}) → (((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ 𝑣 ↔ ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤})))
53	52	rspcev 3541	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∪ {𝑤}) ∈ ω ∧ ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤})) → ∃𝑣 ∈ ω ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ 𝑣)
54	51, 53	sylan 583	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ω ∧ ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤})) → ∃𝑣 ∈ ω ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ 𝑣)
55		isfi 8551	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∪ 𝑦) ∪ {𝑧}) ∈ Fin ↔ ∃𝑣 ∈ ω ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ 𝑣)
56	54, 55	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ω ∧ ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤})) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ∈ Fin)
57	48, 56	eqeltrrid 2857	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤})) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
58	47, 57	syldan 594	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ω ∧ ((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦))) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
59	58	rexlimiva 3205	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ ω ((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
60	29, 59	sylbir 238	. . . . . . . . . 10 ⊢ ((∃𝑤 ∈ ω (𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
61	28, 60	sylan 583	. . . . . . . . 9 ⊢ (((𝐴 ∪ 𝑦) ∈ Fin ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
62	61	ancoms 462	. . . . . . . 8 ⊢ (((¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
63	62	expl 461	. . . . . . 7 ⊢ (¬ 𝑧 ∈ 𝐴 → ((¬ 𝑧 ∈ 𝑦 ∧ (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
64	26, 63	pm2.61i 185	. . . . . 6 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
65	64	ex 416	. . . . 5 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝐴 ∪ 𝑦) ∈ Fin → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
66	65	imim2d 57	. . . 4 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝐴 ∈ Fin → (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∈ Fin → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)))
67	66	adantl 485	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐴 ∈ Fin → (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∈ Fin → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)))
68	3, 6, 9, 12, 15, 67	findcard2s 8736	. 2 ⊢ (𝐵 ∈ Fin → (𝐴 ∈ Fin → (𝐴 ∪ 𝐵) ∈ Fin))
69	68	impcom 411	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∃wrex 3071  Vcvv 3409   ∪ cun 3856   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225  {csn 4522   class class class wbr 5032  Ord word 6168  suc csuc 6171  ωcom 7579   ≈ cen 8524  Fincfn 8527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-en 8528  df-fin 8531

This theorem is referenced by:  ssfi  8742  imafi  8743  pwfilem  8745  unfi2  8820  difinf  8821  xpfi  8822  prfi  8826  tpfi  8827  fnfi  8829  iunfi  8845  pwfilemOLD  8851  fsuppun  8885  fsuppunfi  8886  ressuppfi  8892  fiin  8919  cantnfp1lem1  9174  ficardadju  9659  ficardun2  9662  ficardun2OLD  9663  ackbij1lem6  9685  ackbij1lem16  9695  fin23lem28  9800  fin23lem30  9802  isfin1-3  9846  axcclem  9917  hashun  13793  hashunlei  13836  hashmap  13846  hashbclem  13860  hashf1lem1OLD  13865  hashf1lem2  13866  hashf1  13867  fsumsplitsn  15148  fsummsnunz  15157  fsumsplitsnun  15158  incexclem  15239  isumltss  15251  fprodsplitsn  15391  lcmfunsnlem2lem1  16034  lcmfunsnlem2lem2  16035  lcmfunsnlem2  16036  lcmfun  16041  ramub1lem1  16417  fpwipodrs  17840  acsfiindd  17853  symgfisg  18663  gsumzunsnd  19144  gsumunsnfd  19145  dsmmacl  20506  psrbagaddclOLD  20691  mplsubg  20767  mpllss  20768  fctop  21704  uncmp  22103  bwth  22110  lfinun  22225  locfincmp  22226  comppfsc  22232  1stckgenlem  22253  ptbasin  22277  cfinfil  22593  fin1aufil  22632  alexsubALTlem3  22749  tmdgsum  22795  tsmsfbas  22828  tsmsgsum  22839  tsmsres  22844  tsmsxplem1  22853  prdsmet  23072  prdsbl  23193  icccmplem2  23524  rrxmval  24105  rrxmet  24108  rrxdstprj1  24109  ovolfiniun  24201  volfiniun  24247  fta1glem2  24866  fta1lem  25002  aannenlem2  25024  aalioulem2  25028  dchrfi  25938  usgrfilem  27216  ffsrn  30588  eulerpartlemt  31857  ballotlemgun  32010  hgt750lemb  32155  hgt750leme  32157  lindsenlbs  35332  poimirlem31  35368  poimirlem32  35369  itg2addnclem2  35389  ftc1anclem7  35416  ftc1anc  35418  prdsbnd  35511  pclfinN  37476  elrfi  40008  mzpcompact2lem  40065  eldioph2  40076  lsmfgcl  40391  fiuneneq  40514  unfid  42161  dvmptfprodlem  42952  dvnprodlem2  42955  fourierdlem50  43164  fourierdlem51  43165  fourierdlem54  43168  fourierdlem76  43190  fourierdlem80  43194  fourierdlem102  43216  fourierdlem103  43217  fourierdlem104  43218  fourierdlem114  43228  sge0resplit  43411  sge0iunmptlemfi  43418  sge0xaddlem1  43438  hoiprodp1  43593  sge0hsphoire  43594  hoidmvlelem1  43600  hoidmvlelem2  43601  hoidmvlelem5  43604  hspmbllem2  43632  fsummmodsnunz  44260  mndpsuppfi  45144