Theorem cfslb 10174

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 6845	. . 3 ⊢ (card‘𝐵) ∈ V
2		ssid 3954	. . . . . . 7 ⊢ (card‘𝐵) ⊆ (card‘𝐵)
3		cfslb.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
4	3	ssex 5264	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ∈ V)
5	4	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → 𝐵 ∈ V)
6		velpw 4557	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
7		sseq1 3957	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
8	6, 7	bitrid 283	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
9		unieq 4872	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ∪ 𝑥 = ∪ 𝐵)
10	9	eqeq1d 2736	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (∪ 𝑥 = 𝐴 ↔ ∪ 𝐵 = 𝐴))
11	8, 10	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴)))
12		fveq2 6832	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (card‘𝑥) = (card‘𝐵))
13	12	sseq1d 3963	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((card‘𝑥) ⊆ (card‘𝐵) ↔ (card‘𝐵) ⊆ (card‘𝐵)))
14	11, 13	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵))))
15	14	spcegv 3549	. . . . . . . . 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 3059	. . . . . . . . 9 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)))
18		rabid 3418	. . . . . . . . . . 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 1849	. . . . . . . . 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 275	. . . . . . . 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 234	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
23	2, 22	mpan2 691	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
24		iinss 5010	. . . . . 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 10170	. . . . . 6 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))
27	26	sseq1d 3963	. . . . 5 ⊢ (Lim 𝐴 → ((cf‘𝐴) ⊆ (card‘𝐵) ↔ ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵)))
28	25, 27	imbitrrid 246	. . . 4 ⊢ (Lim 𝐴 → ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵)))
29	28	3impib 1116	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵))
30		ssdomg 8935	. . 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 6376	. . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴)
33		ordsson 7726	. . . . . . 7 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
34	32, 33	syl 17	. . . . . 6 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
35		sstr2 3938	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On))
36	34, 35	mpan9 506	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ On)
37		onssnum 9948	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐵 ⊆ On) → 𝐵 ∈ dom card)
38	4, 36, 37	syl2an2 686	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ dom card)
39		cardid2 9863	. . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
40	38, 39	syl 17	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (card‘𝐵) ≈ 𝐵)
41	40	3adant3 1132	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (card‘𝐵) ≈ 𝐵)
42		domentr 8948	. 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 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3058  {crab 3397  Vcvv 3438   ⊆ wss 3899  𝒫 cpw 4552  ∪ cuni 4861  ∩ ciin 4945   class class class wbr 5096  dom cdm 5622  Ord word 6314  Oncon0 6315  Lim wlim 6316  ‘cfv 6490   ≈ cen 8878   ≼ cdom 8879  cardccrd 9845  cfccf 9847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-er 8633  df-en 8882  df-dom 8883  df-card 9849  df-cf 9851

This theorem is referenced by:  cfslbn  10175  cfslb2n  10176  rankcf  10686