Theorem cflecard 9940

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 9934	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2		df-sn 4559	. . . . . 6 ⊢ {(card‘𝐴)} = {𝑥 ∣ 𝑥 = (card‘𝐴)}
3		ssid 3939	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
4		ssid 3939	. . . . . . . . . . 11 ⊢ 𝑧 ⊆ 𝑧
5		sseq2 3943	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝑧))
6	5	rspcev 3552	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ⊆ 𝑧) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
7	4, 6	mpan2 687	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
8	7	rgen 3073	. . . . . . . . 9 ⊢ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤
9	3, 8	pm3.2i 470	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
10		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (card‘𝑦) = (card‘𝐴))
11	10	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝐴)))
12		sseq1 3942	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
13		rexeq 3334	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
14	13	ralbidv 3120	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
15	12, 14	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)))
16	11, 15	anbi12d 630	. . . . . . . . 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 3526	. . . . . . . 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 693	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝑥 = (card‘𝐴) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
19	18	ss2abdv 3993	. . . . . 6 ⊢ (𝐴 ∈ On → {𝑥 ∣ 𝑥 = (card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
20	2, 19	eqsstrid 3965	. . . . 5 ⊢ (𝐴 ∈ On → {(card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
21		intss 4897	. . . . 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 6769	. . . . 5 ⊢ (card‘𝐴) ∈ V
24	23	intsn 4914	. . . 4 ⊢ ∩ {(card‘𝐴)} = (card‘𝐴)
25	22, 24	sseqtrdi 3967	. . 3 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ (card‘𝐴))
26	1, 25	eqsstrd 3955	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
27		cff 9935	. . . . . 6 ⊢ cf:On⟶On
28	27	fdmi 6596	. . . . 5 ⊢ dom cf = On
29	28	eleq2i 2830	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
30		ndmfv 6786	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
31	29, 30	sylnbir 330	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
32		0ss 4327	. . 3 ⊢ ∅ ⊆ (card‘𝐴)
33	31, 32	eqsstrdi 3971	. 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 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∩ cint 4876  dom cdm 5580  Oncon0 6251  ‘cfv 6418  cardccrd 9624  cfccf 9626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-card 9628  df-cf 9630

This theorem is referenced by:  cfle  9941