Theorem cfslb 10188

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 6855	. . 3 ⊢ (card‘𝐵) ∈ V
2		ssid 3958	. . . . . . 7 ⊢ (card‘𝐵) ⊆ (card‘𝐵)
3		cfslb.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
4	3	ssex 5268	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ∈ V)
5	4	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵)) → 𝐵 ∈ V)
6		velpw 4561	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
7		sseq1 3961	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
8	6, 7	bitrid 283	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
9		unieq 4876	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ∪ 𝑥 = ∪ 𝐵)
10	9	eqeq1d 2739	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (∪ 𝑥 = 𝐴 ↔ ∪ 𝐵 = 𝐴))
11	8, 10	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴)))
12		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (card‘𝑥) = (card‘𝐵))
13	12	sseq1d 3967	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((card‘𝑥) ⊆ (card‘𝐵) ↔ (card‘𝐵) ⊆ (card‘𝐵)))
14	11, 13	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ∧ (card‘𝑥) ⊆ (card‘𝐵)) ↔ ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) ∧ (card‘𝐵) ⊆ (card‘𝐵))))
15	14	spcegv 3553	. . . . . . . . 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 3063	. . . . . . . . 9 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ (card‘𝑥) ⊆ (card‘𝐵)))
18		rabid 3422	. . . . . . . . . . 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 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 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 692	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) ⊆ (card‘𝐵))
24		iinss 5014	. . . . . 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 10184	. . . . . 6 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))
27	26	sseq1d 3967	. . . . 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 1117	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ⊆ (card‘𝐵))
30		ssdomg 8949	. . 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 6386	. . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴)
33		ordsson 7738	. . . . . . 7 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
34	32, 33	syl 17	. . . . . 6 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
35		sstr2 3942	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ⊆ On → 𝐵 ⊆ On))
36	34, 35	mpan9 506	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ On)
37		onssnum 9962	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐵 ⊆ On) → 𝐵 ∈ dom card)
38	4, 36, 37	syl2an2 687	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ dom card)
39		cardid2 9877	. . . 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 8962	. 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 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062  {crab 3401  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865  ∩ ciin 4949   class class class wbr 5100  dom cdm 5632  Ord word 6324  Oncon0 6325  Lim wlim 6326  ‘cfv 6500   ≈ cen 8892   ≼ cdom 8893  cardccrd 9859  cfccf 9861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-er 8645  df-en 8896  df-dom 8897  df-card 9863  df-cf 9865

This theorem is referenced by:  cfslbn  10189  cfslb2n  10190  rankcf  10700