Theorem infssuni 8967

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 3159	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin ↔ ¬ ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)
2		iunfi 8964	. . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ ∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin) → ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin)
3		iunin2 4979	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥)
4	3	eleq1i 2828	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin ↔ (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin)
5		uniiun 4967	. . . . . . . . . . . 12 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
6	5	eqcomi 2746	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ 𝐵 𝑥 = ∪ 𝐵
7	6	ineq2i 4124	. . . . . . . . . 10 ⊢ (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) = (𝐴 ∩ ∪ 𝐵)
8	7	eleq1i 2828	. . . . . . . . 9 ⊢ ((𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin ↔ (𝐴 ∩ ∪ 𝐵) ∈ Fin)
9		df-ss 3883	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ (𝐴 ∩ ∪ 𝐵) = 𝐴)
10		eleq1 2825	. . . . . . . . . . . 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	syl6bi 256	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ ∪ 𝐵) = 𝐴 → ((𝐴 ∩ ∪ 𝐵) ∈ Fin → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)))
13	9, 12	sylbi 220	. . . . . . . . . 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 220	. . . . . . . 8 ⊢ ((𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin → (𝐴 ⊆ ∪ 𝐵 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)))
16	4, 15	sylbi 220	. . . . . . 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 416	. . . . 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 1116	. . 3 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐴 ⊆ ∪ 𝐵) → (∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin))
21	1, 20	syl5bir 246	. 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 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062   ∩ cin 3865   ⊆ wss 3866  ∪ cuni 4819  ∪ ciun 4904  Fincfn 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-en 8627  df-fin 8630

This theorem is referenced by:  bwth  22307