Theorem cfslb 9410

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 6450	. . 3 ⊢ (card‘𝐵) ∈ V
2		ssid 3848	. . . . . . 7 ⊢ (card‘𝐵) ⊆ (card‘𝐵)
3		cfslb.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
4	3	ssex 5029	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ∈ V)
5	4	ad2antrr 717	. . . . . . . . 9 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → 𝐵 ∈ V)
6		selpw 4387	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
7		sseq1 3851	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
8	6, 7	syl5bb 275	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
9		unieq 4668	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ∪ 𝑥 = ∪ 𝐵)
10	9	eqeq1d 2827	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (∪ 𝑥 = 𝐴 ↔ ∪ 𝐵 = 𝐴))
11	8, 10	anbi12d 624	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴)))
12		fveq2 6437	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (card‘𝑥) = (card‘𝐵))
13	12	sseq1d 3857	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((card‘𝑥) ⊆ (card‘𝐵) ↔ (card‘𝐵) ⊆ (card‘𝐵)))
14	11, 13	anbi12d 624	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵))))
15	14	spcegv 3511	. . . . . . . . 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 3123	. . . . . . . . 9 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)))
18		rabid 3326	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
19	18	anbi1i 617	. . . . . . . . . 10 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)))
20	19	exbii 1947	. . . . . . . . 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 267	. . . . . . . 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 226	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
23	2, 22	mpan2 682	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
24		iinss 4793	. . . . . 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 9406	. . . . . 6 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))
27	26	sseq1d 3857	. . . . 5 ⊢ (Lim 𝐴 → ((cf‘𝐴) ⊆ (card‘𝐵) ↔ ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵)))
28	25, 27	syl5ibr 238	. . . 4 ⊢ (Lim 𝐴 → ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵)))
29	28	3impib 1148	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵))
30		ssdomg 8274	. . 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	4	adantl 475	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
33		limord 6026	. . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴)
34		ordsson 7255	. . . . . . 7 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
35	33, 34	syl 17	. . . . . 6 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
36		sstr2 3834	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On))
37	35, 36	mpan9 502	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ On)
38		onssnum 9183	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐵 ⊆ On) → 𝐵 ∈ dom card)
39	32, 37, 38	syl2anc 579	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ dom card)
40		cardid2 9099	. . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
41	39, 40	syl 17	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (card‘𝐵) ≈ 𝐵)
42	41	3adant3 1166	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (card‘𝐵) ≈ 𝐵)
43		domentr 8287	. 2 ⊢ (((cf‘𝐴) ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → (cf‘𝐴) ≼ 𝐵)
44	31, 42, 43	syl2anc 579	1 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656  ∃wex 1878   ∈ wcel 2164  ∃wrex 3118  {crab 3121  Vcvv 3414   ⊆ wss 3798  𝒫 cpw 4380  ∪ cuni 4660  ∩ ciin 4743   class class class wbr 4875  dom cdm 5346  Ord word 5966  Oncon0 5967  Lim wlim 5968  ‘cfv 6127   ≈ cen 8225   ≼ cdom 8226  cardccrd 9081  cfccf 9083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-wrecs 7677  df-recs 7739  df-er 8014  df-en 8229  df-dom 8230  df-card 9085  df-cf 9087

This theorem is referenced by:  cfslbn  9411  cfslb2n  9412  rankcf  9921