Theorem cflecard 10166

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 10160	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2		df-sn 4556	. . . . . 6 ⊢ {(card‘𝐴)} = {𝑥 ∣ 𝑥 = (card‘𝐴)}
3		ssid 3937	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
4		ssid 3937	. . . . . . . . . . 11 ⊢ 𝑧 ⊆ 𝑧
5		sseq2 3941	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝑧))
6	5	rspcev 3560	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ⊆ 𝑧) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
7	4, 6	mpan2 697	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
8	7	rgen 3055	. . . . . . . . 9 ⊢ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤
9	3, 8	pm3.2i 471	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
10		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (card‘𝑦) = (card‘𝐴))
11	10	eqeq2d 2750	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝐴)))
12		sseq1 3940	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
13		rexeq 3293	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
14	13	ralbidv 3162	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
15	12, 14	anbi12d 638	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)))
16	11, 15	anbi12d 638	. . . . . . . . 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 3535	. . . . . . . 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 703	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝑥 = (card‘𝐴) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
19	18	ss2abdv 3996	. . . . . 6 ⊢ (𝐴 ∈ On → {𝑥 ∣ 𝑥 = (card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
20	2, 19	eqsstrid 3953	. . . . 5 ⊢ (𝐴 ∈ On → {(card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
21		intss 4899	. . . . 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 6840	. . . . 5 ⊢ (card‘𝐴) ∈ V
24	23	intsn 4914	. . . 4 ⊢ ∩ {(card‘𝐴)} = (card‘𝐴)
25	22, 24	sseqtrdi 3955	. . 3 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ (card‘𝐴))
26	1, 25	eqsstrd 3949	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
27		cff 10161	. . . . . 6 ⊢ cf:On⟶On
28	27	fdmi 6666	. . . . 5 ⊢ dom cf = On
29	28	eleq2i 2831	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
30		ndmfv 6859	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
31	29, 30	sylnbir 332	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
32		0ss 4328	. . 3 ⊢ ∅ ⊆ (card‘𝐴)
33	31, 32	eqsstrdi 3959	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
34	26, 33	pm2.61i 183	1 ⊢ (cf‘𝐴) ⊆ (card‘𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063   ⊆ wss 3883  ∅c0 4261  {csn 4555  ∩ cint 4877  dom cdm 5618  Oncon0 6310  ‘cfv 6485  cardccrd 9850  cfccf 9852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-card 9854  df-cf 9856

This theorem is referenced by:  cfle  10167