Theorem cflecard 10150

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 10144	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2		df-sn 4576	. . . . . 6 ⊢ {(card‘𝐴)} = {𝑥 ∣ 𝑥 = (card‘𝐴)}
3		ssid 3952	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
4		ssid 3952	. . . . . . . . . . 11 ⊢ 𝑧 ⊆ 𝑧
5		sseq2 3956	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝑧))
6	5	rspcev 3572	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ⊆ 𝑧) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
7	4, 6	mpan2 691	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
8	7	rgen 3049	. . . . . . . . 9 ⊢ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤
9	3, 8	pm3.2i 470	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
10		fveq2 6828	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (card‘𝑦) = (card‘𝐴))
11	10	eqeq2d 2742	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝐴)))
12		sseq1 3955	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
13		rexeq 3288	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))
14	13	ralbidv 3155	. . . . . . . . . . 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 3547	. . . . . . . 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 4013	. . . . . 6 ⊢ (𝐴 ∈ On → {𝑥 ∣ 𝑥 = (card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
20	2, 19	eqsstrid 3968	. . . . 5 ⊢ (𝐴 ∈ On → {(card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
21		intss 4919	. . . . 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 6841	. . . . 5 ⊢ (card‘𝐴) ∈ V
24	23	intsn 4934	. . . 4 ⊢ ∩ {(card‘𝐴)} = (card‘𝐴)
25	22, 24	sseqtrdi 3970	. . 3 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ (card‘𝐴))
26	1, 25	eqsstrd 3964	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
27		cff 10145	. . . . . 6 ⊢ cf:On⟶On
28	27	fdmi 6668	. . . . 5 ⊢ dom cf = On
29	28	eleq2i 2823	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
30		ndmfv 6860	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
31	29, 30	sylnbir 331	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
32		0ss 4349	. . 3 ⊢ ∅ ⊆ (card‘𝐴)
33	31, 32	eqsstrdi 3974	. 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 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  ∅c0 4282  {csn 4575  ∩ cint 4897  dom cdm 5619  Oncon0 6312  ‘cfv 6487  cardccrd 9834  cfccf 9836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6315  df-on 6316  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fv 6495  df-card 9838  df-cf 9840

This theorem is referenced by:  cfle  10151