Theorem infssuni 9230

Description: If an infinite set 𝐴 is included in the underlying set of a finite cover 𝐵, then there exists a set of the cover that contains an infinite number of element of 𝐴. (Contributed by FL, 2-Aug-2009.)

Assertion

Ref	Expression
infssuni	⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐴 ⊆ ∪ 𝐵) → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem infssuni

Step	Hyp	Ref	Expression
1		dfral2 3083	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin ↔ ¬ ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)
2		iunfi 9227	. . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ ∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin) → ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin)
3		iunin2 5019	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥)
4	3	eleq1i 2822	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin ↔ (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin)
5		uniiun 5007	. . . . . . . . . . . 12 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
6	5	eqcomi 2740	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ 𝐵 𝑥 = ∪ 𝐵
7	6	ineq2i 4167	. . . . . . . . . 10 ⊢ (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) = (𝐴 ∩ ∪ 𝐵)
8	7	eleq1i 2822	. . . . . . . . 9 ⊢ ((𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin ↔ (𝐴 ∩ ∪ 𝐵) ∈ Fin)
9		dfss2 3920	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ (𝐴 ∩ ∪ 𝐵) = 𝐴)
10		eleq1 2819	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ ∪ 𝐵) = 𝐴 → ((𝐴 ∩ ∪ 𝐵) ∈ Fin ↔ 𝐴 ∈ Fin))
11		pm2.24 124	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin))
12	10, 11	biimtrdi 253	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ ∪ 𝐵) = 𝐴 → ((𝐴 ∩ ∪ 𝐵) ∈ Fin → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)))
13	9, 12	sylbi 217	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ∪ 𝐵 → ((𝐴 ∩ ∪ 𝐵) ∈ Fin → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)))
14	13	com12 32	. . . . . . . . 9 ⊢ ((𝐴 ∩ ∪ 𝐵) ∈ Fin → (𝐴 ⊆ ∪ 𝐵 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)))
15	8, 14	sylbi 217	. . . . . . . 8 ⊢ ((𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin → (𝐴 ⊆ ∪ 𝐵 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)))
16	4, 15	sylbi 217	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin → (𝐴 ⊆ ∪ 𝐵 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)))
17	2, 16	syl 17	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ ∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin) → (𝐴 ⊆ ∪ 𝐵 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)))
18	17	ex 412	. . . . 5 ⊢ (𝐵 ∈ Fin → (∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin → (𝐴 ⊆ ∪ 𝐵 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin))))
19	18	com24 95	. . . 4 ⊢ (𝐵 ∈ Fin → (¬ 𝐴 ∈ Fin → (𝐴 ⊆ ∪ 𝐵 → (∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin))))
20	19	3imp21 1113	. . 3 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐴 ⊆ ∪ 𝐵) → (∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin))
21	1, 20	biimtrrid 243	. 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐴 ⊆ ∪ 𝐵) → (¬ ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin))
22	21	pm2.18d 127	1 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐴 ⊆ ∪ 𝐵) → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ∩ cin 3901   ⊆ wss 3902  ∪ cuni 4859  ∪ ciun 4941  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 5234  ax-nul 5244  ax-pr 5370  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 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  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:  bwth  23323