Theorem cfslb 10022

Description: Any cofinal subset of 𝐴 is at least as large as (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypothesis

Ref	Expression
cfslb.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
cfslb	⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)

Proof of Theorem cfslb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6787	. . 3 ⊢ (card‘𝐵) ∈ V
2		ssid 3943	. . . . . . 7 ⊢ (card‘𝐵) ⊆ (card‘𝐵)
3		cfslb.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
4	3	ssex 5245	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ∈ V)
5	4	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → 𝐵 ∈ V)
6		velpw 4538	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
7		sseq1 3946	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
8	6, 7	bitrid 282	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
9		unieq 4850	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ∪ 𝑥 = ∪ 𝐵)
10	9	eqeq1d 2740	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (∪ 𝑥 = 𝐴 ↔ ∪ 𝐵 = 𝐴))
11	8, 10	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴)))
12		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (card‘𝑥) = (card‘𝐵))
13	12	sseq1d 3952	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((card‘𝑥) ⊆ (card‘𝐵) ↔ (card‘𝐵) ⊆ (card‘𝐵)))
14	11, 13	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵))))
15	14	spcegv 3536	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵))))
16	5, 15	mpcom 38	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
17		df-rex 3070	. . . . . . . . 9 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)))
18		rabid 3310	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
19	18	anbi1i 624	. . . . . . . . . 10 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
20	19	exbii 1850	. . . . . . . . 9 ⊢ (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
21	17, 20	bitri 274	. . . . . . . 8 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
22	16, 21	sylibr 233	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
23	2, 22	mpan2 688	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
24		iinss 4986	. . . . . 6 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) → ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
25	23, 24	syl 17	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
26	3	cflim3 10018	. . . . . 6 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))
27	26	sseq1d 3952	. . . . 5 ⊢ (Lim 𝐴 → ((cf‘𝐴) ⊆ (card‘𝐵) ↔ ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵)))
28	25, 27	syl5ibr 245	. . . 4 ⊢ (Lim 𝐴 → ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵)))
29	28	3impib 1115	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵))
30		ssdomg 8786	. . 3 ⊢ ((card‘𝐵) ∈ V → ((cf‘𝐴) ⊆ (card‘𝐵) → (cf‘𝐴) ≼ (card‘𝐵)))
31	1, 29, 30	mpsyl 68	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ≼ (card‘𝐵))
32		limord 6325	. . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴)
33		ordsson 7633	. . . . . . 7 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
34	32, 33	syl 17	. . . . . 6 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
35		sstr2 3928	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On))
36	34, 35	mpan9 507	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ On)
37		onssnum 9796	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐵 ⊆ On) → 𝐵 ∈ dom card)
38	4, 36, 37	syl2an2 683	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ dom card)
39		cardid2 9711	. . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
40	38, 39	syl 17	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (card‘𝐵) ≈ 𝐵)
41	40	3adant3 1131	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (card‘𝐵) ≈ 𝐵)
42		domentr 8799	. 2 ⊢ (((cf‘𝐴) ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → (cf‘𝐴) ≼ 𝐵)
43	31, 41, 42	syl2anc 584	1 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065  {crab 3068  Vcvv 3432   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  ∩ ciin 4925   class class class wbr 5074  dom cdm 5589  Ord word 6265  Oncon0 6266  Lim wlim 6267  ‘cfv 6433   ≈ cen 8730   ≼ cdom 8731  cardccrd 9693  cfccf 9695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-er 8498  df-en 8734  df-dom 8735  df-card 9697  df-cf 9699

This theorem is referenced by:  cfslbn  10023  cfslb2n  10024  rankcf  10533