Theorem unfi 9080

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 5301. (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 4109	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ∪ 𝑥) = (𝐴 ∪ ∅))
2	1	eleq1d 2816	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ∪ 𝑥) ∈ Fin ↔ (𝐴 ∪ ∅) ∈ Fin))
3	2	imbi2d 340	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ∈ Fin → (𝐴 ∪ 𝑥) ∈ Fin) ↔ (𝐴 ∈ Fin → (𝐴 ∪ ∅) ∈ Fin)))
4		uneq2 4109	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ∪ 𝑥) = (𝐴 ∪ 𝑦))
5	4	eleq1d 2816	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ∪ 𝑥) ∈ Fin ↔ (𝐴 ∪ 𝑦) ∈ Fin))
6	5	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ Fin → (𝐴 ∪ 𝑥) ∈ Fin) ↔ (𝐴 ∈ Fin → (𝐴 ∪ 𝑦) ∈ Fin)))
7		uneq2 4109	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ∪ 𝑥) = (𝐴 ∪ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2816	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∪ 𝑥) ∈ Fin ↔ (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
9	8	imbi2d 340	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∈ Fin → (𝐴 ∪ 𝑥) ∈ Fin) ↔ (𝐴 ∈ Fin → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)))
10		uneq2 4109	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∪ 𝑥) = (𝐴 ∪ 𝐵))
11	10	eleq1d 2816	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∪ 𝑥) ∈ Fin ↔ (𝐴 ∪ 𝐵) ∈ Fin))
12	11	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ Fin → (𝐴 ∪ 𝑥) ∈ Fin) ↔ (𝐴 ∈ Fin → (𝐴 ∪ 𝐵) ∈ Fin)))
13		un0 4341	. . . . 5 ⊢ (𝐴 ∪ ∅) = 𝐴
14	13	eleq1i 2822	. . . 4 ⊢ ((𝐴 ∪ ∅) ∈ Fin ↔ 𝐴 ∈ Fin)
15	14	biimpri 228	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ∪ ∅) ∈ Fin)
16		snssi 4757	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 → {𝑧} ⊆ 𝐴)
17		ssequn2 4136	. . . . . . . . . . . . . 14 ⊢ ({𝑧} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑧}) = 𝐴)
18	17	biimpi 216	. . . . . . . . . . . . 13 ⊢ ({𝑧} ⊆ 𝐴 → (𝐴 ∪ {𝑧}) = 𝐴)
19	18	uneq2d 4115	. . . . . . . . . . . 12 ⊢ ({𝑧} ⊆ 𝐴 → (𝑦 ∪ (𝐴 ∪ {𝑧})) = (𝑦 ∪ 𝐴))
20		un12 4120	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ (𝑦 ∪ {𝑧})) = (𝑦 ∪ (𝐴 ∪ {𝑧}))
21		uncom 4105	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ 𝑦) = (𝑦 ∪ 𝐴)
22	19, 20, 21	3eqtr4g 2791	. . . . . . . . . . 11 ⊢ ({𝑧} ⊆ 𝐴 → (𝐴 ∪ (𝑦 ∪ {𝑧})) = (𝐴 ∪ 𝑦))
23	16, 22	syl 17	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → (𝐴 ∪ (𝑦 ∪ {𝑧})) = (𝐴 ∪ 𝑦))
24	23	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → ((𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin ↔ (𝐴 ∪ 𝑦) ∈ Fin))
25	24	biimprd 248	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → ((𝐴 ∪ 𝑦) ∈ Fin → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
26	25	adantld 490	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((¬ 𝑧 ∈ 𝑦 ∧ (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
27		isfi 8898	. . . . . . . . . . 11 ⊢ ((𝐴 ∪ 𝑦) ∈ Fin ↔ ∃𝑤 ∈ ω (𝐴 ∪ 𝑦) ≈ 𝑤)
28	27	biimpi 216	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝑦) ∈ Fin → ∃𝑤 ∈ ω (𝐴 ∪ 𝑦) ≈ 𝑤)
29		r19.41v 3162	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ ω ((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)) ↔ (∃𝑤 ∈ ω (𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)))
30		disjsn 4661	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∪ 𝑦) ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ (𝐴 ∪ 𝑦))
31		elun 4100	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝐴 ∪ 𝑦) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
32	31	notbii 320	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑧 ∈ (𝐴 ∪ 𝑦) ↔ ¬ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
33		pm4.56 990	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦) ↔ ¬ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
34	32, 33	bitr4i 278	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑧 ∈ (𝐴 ∪ 𝑦) ↔ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦))
35	30, 34	sylbbr 236	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐴 ∪ 𝑦) ∩ {𝑧}) = ∅)
36		nnord 7804	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ω → Ord 𝑤)
37		orddisj 6344	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑤 → (𝑤 ∩ {𝑤}) = ∅)
38	36, 37	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ω → (𝑤 ∩ {𝑤}) = ∅)
39		en2sn 8963	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → {𝑧} ≈ {𝑤})
40	39	el2v 3443	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑧} ≈ {𝑤}
41		unen 8967	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ {𝑧} ≈ {𝑤}) ∧ (((𝐴 ∪ 𝑦) ∩ {𝑧}) = ∅ ∧ (𝑤 ∩ {𝑤}) = ∅)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
42	40, 41	mpanl2 701	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (((𝐴 ∪ 𝑦) ∩ {𝑧}) = ∅ ∧ (𝑤 ∩ {𝑤}) = ∅)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
43	38, 42	sylanr2 683	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (((𝐴 ∪ 𝑦) ∩ {𝑧}) = ∅ ∧ 𝑤 ∈ ω)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
44	35, 43	sylanr1 682	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ ((¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑤 ∈ ω)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
45	44	3impb 1114	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑤 ∈ ω) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
46	45	3comr 1125	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ω ∧ (𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
47	46	3expb 1120	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ ((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦))) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤}))
48		unass 4119	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∪ 𝑦) ∪ {𝑧}) = (𝐴 ∪ (𝑦 ∪ {𝑧}))
49		df-suc 6312	. . . . . . . . . . . . . . . . 17 ⊢ suc 𝑤 = (𝑤 ∪ {𝑤})
50		peano2 7820	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ω → suc 𝑤 ∈ ω)
51	49, 50	eqeltrrid 2836	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ω → (𝑤 ∪ {𝑤}) ∈ ω)
52		breq2 5093	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑤 ∪ {𝑤}) → (((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ 𝑣 ↔ ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤})))
53	52	rspcev 3572	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∪ {𝑤}) ∈ ω ∧ ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤})) → ∃𝑣 ∈ ω ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ 𝑣)
54	51, 53	sylan 580	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ω ∧ ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤})) → ∃𝑣 ∈ ω ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ 𝑣)
55		isfi 8898	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∪ 𝑦) ∪ {𝑧}) ∈ Fin ↔ ∃𝑣 ∈ ω ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ 𝑣)
56	54, 55	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ω ∧ ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤})) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ∈ Fin)
57	48, 56	eqeltrrid 2836	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ ((𝐴 ∪ 𝑦) ∪ {𝑧}) ≈ (𝑤 ∪ {𝑤})) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
58	47, 57	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ω ∧ ((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦))) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
59	58	rexlimiva 3125	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ ω ((𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
60	29, 59	sylbir 235	. . . . . . . . . 10 ⊢ ((∃𝑤 ∈ ω (𝐴 ∪ 𝑦) ≈ 𝑤 ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
61	28, 60	sylan 580	. . . . . . . . 9 ⊢ (((𝐴 ∪ 𝑦) ∈ Fin ∧ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
62	61	ancoms 458	. . . . . . . 8 ⊢ (((¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
63	62	expl 457	. . . . . . 7 ⊢ (¬ 𝑧 ∈ 𝐴 → ((¬ 𝑧 ∈ 𝑦 ∧ (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
64	26, 63	pm2.61i 182	. . . . . 6 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
65	64	ex 412	. . . . 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 481	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐴 ∈ Fin → (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∈ Fin → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)))
68	3, 6, 9, 12, 15, 67	findcard2s 9075	. 2 ⊢ (𝐵 ∈ Fin → (𝐴 ∈ Fin → (𝐴 ∪ 𝐵) ∈ Fin))
69	68	impcom 407	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897  ∅c0 4280  {csn 4573   class class class wbr 5089  Ord word 6305  suc csuc 6308  ωcom 7796   ≈ cen 8866  Fincfn 8869

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-en 8870  df-fin 8873

This theorem is referenced by:  unfid  9081  ssfi  9082  cnvfi  9085  fnfi  9087  unfib  9193  unfi2  9194  difinf  9195  imafiOLD  9200  pwfilem  9202  xpfi  9204  prfiALT  9209  tpfi  9210  fodomfir  9212  iunfi  9227  fsuppun  9271  fsuppunfi  9272  ressuppfi  9279  fiin  9306  cantnfp1lem1  9568  ficardadju  10091  ficardun2  10093  ackbij1lem6  10115  ackbij1lem16  10125  fin23lem28  10231  fin23lem30  10233  isfin1-3  10277  axcclem  10348  hashun  14289  hashunlei  14332  hashmap  14342  hashbclem  14359  hashf1lem2  14363  hashf1  14364  hash7g  14393  s7f1o  14873  fsumsplitsn  15651  fsummsnunz  15661  fsumsplitsnun  15662  incexclem  15743  isumltss  15755  fprodsplitsn  15896  lcmfunsnlem2lem1  16549  lcmfunsnlem2lem2  16550  lcmfunsnlem2  16551  lcmfun  16556  ramub1lem1  16938  fpwipodrs  18446  acsfiindd  18459  mndpsuppfi  18674  symgfisg  19380  gsumzunsnd  19868  gsumunsnfd  19869  dsmmacl  21678  mplsubg  21939  mpllss  21940  fctop  22919  uncmp  23318  bwth  23325  lfinun  23440  locfincmp  23441  comppfsc  23447  1stckgenlem  23468  ptbasin  23492  cfinfil  23808  fin1aufil  23847  alexsubALTlem3  23964  tmdgsum  24010  tsmsfbas  24043  tsmsgsum  24054  tsmsres  24059  tsmsxplem1  24068  prdsmet  24285  prdsbl  24406  icccmplem2  24739  rrxmval  25332  rrxmet  25335  rrxdstprj1  25336  ovolfiniun  25429  volfiniun  25475  fta1glem2  26101  fta1lem  26242  aannenlem2  26264  aalioulem2  26268  dchrfi  27193  usgrfilem  29305  ffsrn  32711  eulerpartlemt  34384  ballotlemgun  34538  hgt750lemb  34669  hgt750leme  34671  lindsenlbs  37663  poimirlem31  37699  poimirlem32  37700  itg2addnclem2  37720  ftc1anclem7  37747  ftc1anc  37749  prdsbnd  37841  pclfinN  39947  elrfi  42735  mzpcompact2lem  42792  eldioph2  42803  lsmfgcl  43115  fiuneneq  43233  dvmptfprodlem  45990  dvnprodlem2  45993  fourierdlem50  46202  fourierdlem51  46203  fourierdlem54  46206  fourierdlem76  46228  fourierdlem80  46232  fourierdlem102  46254  fourierdlem103  46255  fourierdlem104  46256  fourierdlem114  46266  sge0resplit  46452  sge0iunmptlemfi  46459  sge0xaddlem1  46479  hoiprodp1  46634  sge0hsphoire  46635  hoidmvlelem1  46641  hoidmvlelem2  46642  hoidmvlelem5  46645  hspmbllem2  46673  fsummmodsnunz  47414