Theorem infssuni 9249

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 3089	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin ↔ ¬ ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)
2		iunfi 9246	. . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ ∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin) → ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin)
3		iunin2 5014	. . . . . . . . 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 5002	. . . . . . . . . . . 12 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
6	5	eqcomi 2746	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ 𝐵 𝑥 = ∪ 𝐵
7	6	ineq2i 4158	. . . . . . . . . 10 ⊢ (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) = (𝐴 ∩ ∪ 𝐵)
8	7	eleq1i 2828	. . . . . . . . 9 ⊢ ((𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin ↔ (𝐴 ∩ ∪ 𝐵) ∈ Fin)
9		dfss2 3908	. . . . . . . . . . 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	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 1114	. . 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  ∪ cuni 4851  ∪ ciun 4934  Fincfn 8886

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7811  df-en 8887  df-fin 8890

This theorem is referenced by:  bwth  23385