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Theorem cfss 9125
Description: There is a cofinal subset of 𝐴 of cardinality (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfss.1 𝐴 ∈ V
Assertion
Ref Expression
cfss (Lim 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem cfss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cfss.1 . . . . . 6 𝐴 ∈ V
21cflim3 9122 . . . . 5 (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
3 fvex 6239 . . . . . . 7 (card‘𝑥) ∈ V
43dfiin2 4587 . . . . . 6 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
5 cardon 8808 . . . . . . . . . 10 (card‘𝑥) ∈ On
6 eleq1 2718 . . . . . . . . . 10 (𝑦 = (card‘𝑥) → (𝑦 ∈ On ↔ (card‘𝑥) ∈ On))
75, 6mpbiri 248 . . . . . . . . 9 (𝑦 = (card‘𝑥) → 𝑦 ∈ On)
87rexlimivw 3058 . . . . . . . 8 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) → 𝑦 ∈ On)
98abssi 3710 . . . . . . 7 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On
10 limuni 5823 . . . . . . . . . . . 12 (Lim 𝐴𝐴 = 𝐴)
1110eqcomd 2657 . . . . . . . . . . 11 (Lim 𝐴 𝐴 = 𝐴)
12 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
1312eqcomd 2657 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (card‘𝐴) = (card‘𝑥))
1413biantrud 527 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ( 𝐴 = 𝐴 ↔ ( 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥))))
15 unieq 4476 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴 𝑥 = 𝐴)
1615eqeq1d 2653 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → ( 𝑥 = 𝐴 𝐴 = 𝐴))
171pwid 4207 . . . . . . . . . . . . . . . . 17 𝐴 ∈ 𝒫 𝐴
18 eleq1 2718 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐴𝐴 ∈ 𝒫 𝐴))
1917, 18mpbiri 248 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴𝑥 ∈ 𝒫 𝐴)
2019biantrurd 528 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → ( 𝑥 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴)))
2116, 20bitr3d 270 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → ( 𝐴 = 𝐴 ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴)))
2221anbi1d 741 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → (( 𝐴 = 𝐴 ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥))))
2314, 22bitr2d 269 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)) ↔ 𝐴 = 𝐴))
241, 23spcev 3331 . . . . . . . . . . 11 ( 𝐴 = 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
2511, 24syl 17 . . . . . . . . . 10 (Lim 𝐴 → ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
26 df-rex 2947 . . . . . . . . . . 11 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)))
27 rabid 3145 . . . . . . . . . . . . 13 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
2827anbi1i 731 . . . . . . . . . . . 12 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
2928exbii 1814 . . . . . . . . . . 11 (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (card‘𝐴) = (card‘𝑥)) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
3026, 29bitri 264 . . . . . . . . . 10 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) ↔ ∃𝑥((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ∧ (card‘𝐴) = (card‘𝑥)))
3125, 30sylibr 224 . . . . . . . . 9 (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥))
32 fvex 6239 . . . . . . . . . 10 (card‘𝐴) ∈ V
33 eqeq1 2655 . . . . . . . . . . 11 (𝑦 = (card‘𝐴) → (𝑦 = (card‘𝑥) ↔ (card‘𝐴) = (card‘𝑥)))
3433rexbidv 3081 . . . . . . . . . 10 (𝑦 = (card‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥)))
3532, 34spcev 3331 . . . . . . . . 9 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝐴) = (card‘𝑥) → ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
3631, 35syl 17 . . . . . . . 8 (Lim 𝐴 → ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
37 abn0 3987 . . . . . . . 8 ({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅ ↔ ∃𝑦𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥))
3836, 37sylibr 224 . . . . . . 7 (Lim 𝐴 → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅)
39 onint 7037 . . . . . . 7 (({𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ⊆ On ∧ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ≠ ∅) → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
409, 38, 39sylancr 696 . . . . . 6 (Lim 𝐴 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
414, 40syl5eqel 2734 . . . . 5 (Lim 𝐴 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
422, 41eqeltrd 2730 . . . 4 (Lim 𝐴 → (cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)})
43 fvex 6239 . . . . 5 (cf‘𝐴) ∈ V
44 eqeq1 2655 . . . . . 6 (𝑦 = (cf‘𝐴) → (𝑦 = (card‘𝑥) ↔ (cf‘𝐴) = (card‘𝑥)))
4544rexbidv 3081 . . . . 5 (𝑦 = (cf‘𝐴) → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥)))
4643, 45elab 3382 . . . 4 ((cf‘𝐴) ∈ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} ↔ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
4742, 46sylib 208 . . 3 (Lim 𝐴 → ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥))
48 df-rex 2947 . . 3 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (cf‘𝐴) = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
4947, 48sylib 208 . 2 (Lim 𝐴 → ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)))
50 simprl 809 . . . . . . . 8 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴})
5150, 27sylib 208 . . . . . . 7 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
5251simpld 474 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ∈ 𝒫 𝐴)
5352elpwid 4203 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥𝐴)
54 simpl 472 . . . . . . 7 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → Lim 𝐴)
55 vex 3234 . . . . . . . . . 10 𝑥 ∈ V
56 limord 5822 . . . . . . . . . . . 12 (Lim 𝐴 → Ord 𝐴)
57 ordsson 7031 . . . . . . . . . . . 12 (Ord 𝐴𝐴 ⊆ On)
5856, 57syl 17 . . . . . . . . . . 11 (Lim 𝐴𝐴 ⊆ On)
59 sstr 3644 . . . . . . . . . . 11 ((𝑥𝐴𝐴 ⊆ On) → 𝑥 ⊆ On)
6058, 59sylan2 490 . . . . . . . . . 10 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ⊆ On)
61 onssnum 8901 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑥 ⊆ On) → 𝑥 ∈ dom card)
6255, 60, 61sylancr 696 . . . . . . . . 9 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ∈ dom card)
63 cardid2 8817 . . . . . . . . 9 (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
6462, 63syl 17 . . . . . . . 8 ((𝑥𝐴 ∧ Lim 𝐴) → (card‘𝑥) ≈ 𝑥)
6564ensymd 8048 . . . . . . 7 ((𝑥𝐴 ∧ Lim 𝐴) → 𝑥 ≈ (card‘𝑥))
6653, 54, 65syl2anc 694 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (card‘𝑥))
67 simprr 811 . . . . . 6 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (cf‘𝐴) = (card‘𝑥))
6866, 67breqtrrd 4713 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 ≈ (cf‘𝐴))
6951simprd 478 . . . . 5 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → 𝑥 = 𝐴)
7053, 68, 693jca 1261 . . . 4 ((Lim 𝐴 ∧ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥))) → (𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
7170ex 449 . . 3 (Lim 𝐴 → ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → (𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴)))
7271eximdv 1886 . 2 (Lim 𝐴 → (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ (cf‘𝐴) = (card‘𝑥)) → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴)))
7349, 72mpd 15 1 (Lim 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468   cint 4507   ciin 4553   class class class wbr 4685  dom cdm 5143  Ord word 5760  Oncon0 5761  Lim wlim 5762  cfv 5926  cen 7994  cardccrd 8799  cfccf 8801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-wrecs 7452  df-recs 7513  df-er 7787  df-en 7998  df-dom 7999  df-card 8803  df-cf 8805
This theorem is referenced by: (None)
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