Theorem infssuni 8608

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 3177	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin ↔ ¬ ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)
2		iunfi 8605	. . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ ∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin) → ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin)
3		iunin2 4855	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥)
4	3	eleq1i 2849	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin ↔ (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin)
5		uniiun 4844	. . . . . . . . . . . 12 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
6	5	eqcomi 2780	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ 𝐵 𝑥 = ∪ 𝐵
7	6	ineq2i 4067	. . . . . . . . . 10 ⊢ (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) = (𝐴 ∩ ∪ 𝐵)
8	7	eleq1i 2849	. . . . . . . . 9 ⊢ ((𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin ↔ (𝐴 ∩ ∪ 𝐵) ∈ Fin)
9		df-ss 3836	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ (𝐴 ∩ ∪ 𝐵) = 𝐴)
10		eleq1 2846	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ ∪ 𝐵) = 𝐴 → ((𝐴 ∩ ∪ 𝐵) ∈ Fin ↔ 𝐴 ∈ Fin))
11		pm2.24 122	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin))
12	10, 11	syl6bi 245	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ ∪ 𝐵) = 𝐴 → ((𝐴 ∩ ∪ 𝐵) ∈ Fin → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)))
13	9, 12	sylbi 209	. . . . . . . . . 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 209	. . . . . . . 8 ⊢ ((𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥) ∈ Fin → (𝐴 ⊆ ∪ 𝐵 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin)))
16	4, 15	sylbi 209	. . . . . . 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 405	. . . . 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 1095	. . 3 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐴 ⊆ ∪ 𝐵) → (∀𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) ∈ Fin → ∃𝑥 ∈ 𝐵 ¬ (𝐴 ∩ 𝑥) ∈ Fin))
21	1, 20	syl5bir 235	. 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 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051  ∀wral 3081  ∃wrex 3082   ∩ cin 3821   ⊆ wss 3822  ∪ cuni 4708  ∪ ciun 4788  Fincfn 8304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-er 8087  df-en 8305  df-fin 8308

This theorem is referenced by:  bwth  21737