Theorem infssuni 9255

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 3080	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin ↔ ¬ ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)
2		iunfi 9252	. . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ ∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin) → ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin)
3		iunin2 5023	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥)
4	3	eleq1i 2819	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin ↔ (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin)
5		uniiun 5010	. . . . . . . . . . . 12 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
6	5	eqcomi 2738	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ 𝐵 𝑥 = ∪ 𝐵
7	6	ineq2i 4170	. . . . . . . . . 10 ⊢ (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) = (𝐴 ∩ ∪ 𝐵)
8	7	eleq1i 2819	. . . . . . . . 9 ⊢ ((𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin ↔ (𝐴 ∩ ∪ 𝐵) ∈ Fin)
9		dfss2 3923	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ (𝐴 ∩ ∪ 𝐵) = 𝐴)
10		eleq1 2816	. . . . . . . . . . . 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 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3904   ⊆ wss 3905  ∪ cuni 4861  ∪ ciun 4944  Fincfn 8879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-om 7807  df-en 8880  df-fin 8883

This theorem is referenced by:  bwth  23313