Theorem cfle 9278

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 9277	. . 3 ⊢ (cf‘𝐴) ⊆ (card‘𝐴)
2		cardonle 8983	. . 3 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
3	1, 2	syl5ss 3763	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
4		cff 9272	. . . . . 6 ⊢ cf:On⟶On
5	4	fdmi 6192	. . . . 5 ⊢ dom cf = On
6	5	eleq2i 2842	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7		ndmfv 6359	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
8	6, 7	sylnbir 320	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
9		0ss 4116	. . 3 ⊢ ∅ ⊆ 𝐴
10	8, 9	syl6eqss 3804	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
11	3, 10	pm2.61i 176	1 ⊢ (cf‘𝐴) ⊆ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1631   ∈ wcel 2145   ⊆ wss 3723  ∅c0 4063  dom cdm 5249  Oncon0 5866  ‘cfv 6031  cardccrd 8961  cfccf 8963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-en 8110  df-card 8965  df-cf 8967

This theorem is referenced by:  cfom  9288  cfidm  9299  alephreg  9606  winafp  9721  tskcard  9805  gruina  9842