Theorem cfle 10103

Description: Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102. (Contributed by NM, 26-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Assertion

Ref	Expression
cfle	⊢ (cf‘𝐴) ⊆ 𝐴

Proof of Theorem cfle

Step	Hyp	Ref	Expression
1		cflecard 10102	. . 3 ⊢ (cf‘𝐴) ⊆ (card‘𝐴)
2		cardonle 9806	. . 3 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
3	1, 2	sstrid 3942	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
4		cff 10097	. . . . . 6 ⊢ cf:On⟶On
5	4	fdmi 6657	. . . . 5 ⊢ dom cf = On
6	5	eleq2i 2828	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7		ndmfv 6854	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
8	6, 7	sylnbir 330	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
9		0ss 4342	. . 3 ⊢ ∅ ⊆ 𝐴
10	8, 9	eqsstrdi 3985	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
11	3, 10	pm2.61i 182	1 ⊢ (cf‘𝐴) ⊆ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2105   ⊆ wss 3897  ∅c0 4268  dom cdm 5614  Oncon0 6296  ‘cfv 6473  cardccrd 9784  cfccf 9786

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6299  df-on 6300  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-en 8797  df-card 9788  df-cf 9790

This theorem is referenced by:  cfom  10113  cfidm  10124  alephreg  10431  winafp  10546  tskcard  10630  gruina  10667