Theorem cfslb 10257

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 6901	. . 3 ⊢ (card‘𝐵) ∈ V
2		ssid 4003	. . . . . . 7 ⊢ (card‘𝐵) ⊆ (card‘𝐵)
3		cfslb.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
4	3	ssex 5320	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ∈ V)
5	4	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → 𝐵 ∈ V)
6		velpw 4606	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
7		sseq1 4006	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
8	6, 7	bitrid 283	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
9		unieq 4918	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ∪ 𝑥 = ∪ 𝐵)
10	9	eqeq1d 2735	. . . . . . . . . . . 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 6888	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (card‘𝑥) = (card‘𝐵))
13	12	sseq1d 4012	. . . . . . . . . . 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 3587	. . . . . . . . 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 3072	. . . . . . . . 9 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)))
18		rabid 3453	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
19	18	anbi1i 625	. . . . . . . . . 10 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
20	19	exbii 1851	. . . . . . . . 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 233	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
23	2, 22	mpan2 690	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
24		iinss 5058	. . . . . 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 10253	. . . . . 6 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))
27	26	sseq1d 4012	. . . . 5 ⊢ (Lim 𝐴 → ((cf‘𝐴) ⊆ (card‘𝐵) ↔ ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵)))
28	25, 27	imbitrrid 245	. . . 4 ⊢ (Lim 𝐴 → ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵)))
29	28	3impib 1117	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵))
30		ssdomg 8992	. . 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 6421	. . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴)
33		ordsson 7765	. . . . . . 7 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
34	32, 33	syl 17	. . . . . 6 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
35		sstr2 3988	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On))
36	34, 35	mpan9 508	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ On)
37		onssnum 10031	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐵 ⊆ On) → 𝐵 ∈ dom card)
38	4, 36, 37	syl2an2 685	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ dom card)
39		cardid2 9944	. . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
40	38, 39	syl 17	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (card‘𝐵) ≈ 𝐵)
41	40	3adant3 1133	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (card‘𝐵) ≈ 𝐵)
42		domentr 9005	. 2 ⊢ (((cf‘𝐴) ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → (cf‘𝐴) ≼ 𝐵)
43	31, 41, 42	syl2anc 585	1 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∃wrex 3071  {crab 3433  Vcvv 3475   ⊆ wss 3947  𝒫 cpw 4601  ∪ cuni 4907  ∩ ciin 4997   class class class wbr 5147  dom cdm 5675  Ord word 6360  Oncon0 6361  Lim wlim 6362  ‘cfv 6540   ≈ cen 8932   ≼ cdom 8933  cardccrd 9926  cfccf 9928

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-er 8699  df-en 8936  df-dom 8937  df-card 9930  df-cf 9932

This theorem is referenced by:  cfslbn  10258  cfslb2n  10259  rankcf  10768