Theorem cflecard 10267

Description: Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Assertion

Ref	Expression
cflecard	⊢ (cf‘𝐴) ⊆ (card‘𝐴)

Proof of Theorem cflecard

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfval 10261	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2		df-sn 4602	. . . . . 6 ⊢ {(card‘𝐴)} = {𝑥 ∣ 𝑥 = (card‘𝐴)}
3		ssid 3981	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
4		ssid 3981	. . . . . . . . . . 11 ⊢ 𝑧 ⊆ 𝑧
5		sseq2 3985	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝑧))
6	5	rspcev 3601	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ⊆ 𝑧) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
7	4, 6	mpan2 691	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
8	7	rgen 3053	. . . . . . . . 9 ⊢ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤
9	3, 8	pm3.2i 470	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
10		fveq2 6876	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (card‘𝑦) = (card‘𝐴))
11	10	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝐴)))
12		sseq1 3984	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
13		rexeq 3301	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
14	13	ralbidv 3163	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
15	12, 14	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)))
16	11, 15	anbi12d 632	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (𝑥 = (card‘𝐴) ∧ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))))
17	16	spcegv 3576	. . . . . . . 8 ⊢ (𝐴 ∈ On → ((𝑥 = (card‘𝐴) ∧ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
18	9, 17	mpan2i 697	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝑥 = (card‘𝐴) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
19	18	ss2abdv 4041	. . . . . 6 ⊢ (𝐴 ∈ On → {𝑥 ∣ 𝑥 = (card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
20	2, 19	eqsstrid 3997	. . . . 5 ⊢ (𝐴 ∈ On → {(card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
21		intss 4945	. . . . 5 ⊢ ({(card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ ∩ {(card‘𝐴)})
22	20, 21	syl 17	. . . 4 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ ∩ {(card‘𝐴)})
23		fvex 6889	. . . . 5 ⊢ (card‘𝐴) ∈ V
24	23	intsn 4960	. . . 4 ⊢ ∩ {(card‘𝐴)} = (card‘𝐴)
25	22, 24	sseqtrdi 3999	. . 3 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ (card‘𝐴))
26	1, 25	eqsstrd 3993	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
27		cff 10262	. . . . . 6 ⊢ cf:On⟶On
28	27	fdmi 6717	. . . . 5 ⊢ dom cf = On
29	28	eleq2i 2826	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
30		ndmfv 6911	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
31	29, 30	sylnbir 331	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
32		0ss 4375	. . 3 ⊢ ∅ ⊆ (card‘𝐴)
33	31, 32	eqsstrdi 4003	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
34	26, 33	pm2.61i 182	1 ⊢ (cf‘𝐴) ⊆ (card‘𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2713  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  ∅c0 4308  {csn 4601  ∩ cint 4922  dom cdm 5654  Oncon0 6352  ‘cfv 6531  cardccrd 9949  cfccf 9951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-card 9953  df-cf 9955

This theorem is referenced by:  cfle  10268