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Theorem cfslb2n 9128
Description: Any small collection of small subsets of 𝐴 cannot have union 𝐴, where "small" means smaller than the cofinality. This is a stronger version of cfslb 9126. This is a common application of cofinality: under AC, (ℵ‘1) is regular, so it is not a countable union of countable sets. (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfslb.1 𝐴 ∈ V
Assertion
Ref Expression
cfslb2n ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → 𝐵𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem cfslb2n
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 limord 5822 . . . . . . . . . 10 (Lim 𝐴 → Ord 𝐴)
2 ordsson 7031 . . . . . . . . . 10 (Ord 𝐴𝐴 ⊆ On)
3 sstr 3644 . . . . . . . . . . 11 ((𝑥𝐴𝐴 ⊆ On) → 𝑥 ⊆ On)
43expcom 450 . . . . . . . . . 10 (𝐴 ⊆ On → (𝑥𝐴𝑥 ⊆ On))
51, 2, 43syl 18 . . . . . . . . 9 (Lim 𝐴 → (𝑥𝐴𝑥 ⊆ On))
6 onsucuni 7070 . . . . . . . . 9 (𝑥 ⊆ On → 𝑥 ⊆ suc 𝑥)
75, 6syl6 35 . . . . . . . 8 (Lim 𝐴 → (𝑥𝐴𝑥 ⊆ suc 𝑥))
87adantrd 483 . . . . . . 7 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥 ⊆ suc 𝑥))
98ralimdv 2992 . . . . . 6 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 𝑥 ⊆ suc 𝑥))
10 uniiun 4605 . . . . . . 7 𝐵 = 𝑥𝐵 𝑥
11 ss2iun 4568 . . . . . . 7 (∀𝑥𝐵 𝑥 ⊆ suc 𝑥 𝑥𝐵 𝑥 𝑥𝐵 suc 𝑥)
1210, 11syl5eqss 3682 . . . . . 6 (∀𝑥𝐵 𝑥 ⊆ suc 𝑥 𝐵 𝑥𝐵 suc 𝑥)
139, 12syl6 35 . . . . 5 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝐵 𝑥𝐵 suc 𝑥))
1413imp 444 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → 𝐵 𝑥𝐵 suc 𝑥)
15 cfslb.1 . . . . . . . . . 10 𝐴 ∈ V
1615cfslbn 9127 . . . . . . . . 9 ((Lim 𝐴𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐴)
17163expib 1287 . . . . . . . 8 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐴))
18 ordsucss 7060 . . . . . . . 8 (Ord 𝐴 → ( 𝑥𝐴 → suc 𝑥𝐴))
191, 17, 18sylsyld 61 . . . . . . 7 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → suc 𝑥𝐴))
2019ralimdv 2992 . . . . . 6 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 suc 𝑥𝐴))
21 iunss 4593 . . . . . 6 ( 𝑥𝐵 suc 𝑥𝐴 ↔ ∀𝑥𝐵 suc 𝑥𝐴)
2220, 21syl6ibr 242 . . . . 5 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐵 suc 𝑥𝐴))
2322imp 444 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → 𝑥𝐵 suc 𝑥𝐴)
24 sseq1 3659 . . . . . 6 ( 𝐵 = 𝐴 → ( 𝐵 𝑥𝐵 suc 𝑥𝐴 𝑥𝐵 suc 𝑥))
25 eqss 3651 . . . . . . 7 ( 𝑥𝐵 suc 𝑥 = 𝐴 ↔ ( 𝑥𝐵 suc 𝑥𝐴𝐴 𝑥𝐵 suc 𝑥))
2625simplbi2com 656 . . . . . 6 (𝐴 𝑥𝐵 suc 𝑥 → ( 𝑥𝐵 suc 𝑥𝐴 𝑥𝐵 suc 𝑥 = 𝐴))
2724, 26syl6bi 243 . . . . 5 ( 𝐵 = 𝐴 → ( 𝐵 𝑥𝐵 suc 𝑥 → ( 𝑥𝐵 suc 𝑥𝐴 𝑥𝐵 suc 𝑥 = 𝐴)))
2827com3l 89 . . . 4 ( 𝐵 𝑥𝐵 suc 𝑥 → ( 𝑥𝐵 suc 𝑥𝐴 → ( 𝐵 = 𝐴 𝑥𝐵 suc 𝑥 = 𝐴)))
2914, 23, 28sylc 65 . . 3 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ( 𝐵 = 𝐴 𝑥𝐵 suc 𝑥 = 𝐴))
30 limsuc 7091 . . . . . . . . 9 (Lim 𝐴 → ( 𝑥𝐴 ↔ suc 𝑥𝐴))
3117, 30sylibd 229 . . . . . . . 8 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → suc 𝑥𝐴))
3231ralimdv 2992 . . . . . . 7 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 suc 𝑥𝐴))
3332imp 444 . . . . . 6 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ∀𝑥𝐵 suc 𝑥𝐴)
34 r19.29 3101 . . . . . . . 8 ((∀𝑥𝐵 suc 𝑥𝐴 ∧ ∃𝑥𝐵 𝑦 = suc 𝑥) → ∃𝑥𝐵 (suc 𝑥𝐴𝑦 = suc 𝑥))
35 eleq1 2718 . . . . . . . . . 10 (𝑦 = suc 𝑥 → (𝑦𝐴 ↔ suc 𝑥𝐴))
3635biimparc 503 . . . . . . . . 9 ((suc 𝑥𝐴𝑦 = suc 𝑥) → 𝑦𝐴)
3736rexlimivw 3058 . . . . . . . 8 (∃𝑥𝐵 (suc 𝑥𝐴𝑦 = suc 𝑥) → 𝑦𝐴)
3834, 37syl 17 . . . . . . 7 ((∀𝑥𝐵 suc 𝑥𝐴 ∧ ∃𝑥𝐵 𝑦 = suc 𝑥) → 𝑦𝐴)
3938ex 449 . . . . . 6 (∀𝑥𝐵 suc 𝑥𝐴 → (∃𝑥𝐵 𝑦 = suc 𝑥𝑦𝐴))
4033, 39syl 17 . . . . 5 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → (∃𝑥𝐵 𝑦 = suc 𝑥𝑦𝐴))
4140abssdv 3709 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴)
42 vuniex 6996 . . . . . . . 8 𝑥 ∈ V
4342sucex 7053 . . . . . . 7 suc 𝑥 ∈ V
4443dfiun2 4586 . . . . . 6 𝑥𝐵 suc 𝑥 = {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}
4544eqeq1i 2656 . . . . 5 ( 𝑥𝐵 suc 𝑥 = 𝐴 {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴)
4615cfslb 9126 . . . . . 6 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴 {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴) → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥})
47463expia 1286 . . . . 5 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴) → ( {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
4845, 47syl5bi 232 . . . 4 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴) → ( 𝑥𝐵 suc 𝑥 = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
4941, 48syldan 486 . . 3 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ( 𝑥𝐵 suc 𝑥 = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
50 eqid 2651 . . . . . . . . 9 (𝑥𝐵 ↦ suc 𝑥) = (𝑥𝐵 ↦ suc 𝑥)
5150rnmpt 5403 . . . . . . . 8 ran (𝑥𝐵 ↦ suc 𝑥) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}
5243, 50fnmpti 6060 . . . . . . . . . 10 (𝑥𝐵 ↦ suc 𝑥) Fn 𝐵
53 dffn4 6159 . . . . . . . . . 10 ((𝑥𝐵 ↦ suc 𝑥) Fn 𝐵 ↔ (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥))
5452, 53mpbi 220 . . . . . . . . 9 (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥)
55 relsdom 8004 . . . . . . . . . . 11 Rel ≺
5655brrelexi 5192 . . . . . . . . . 10 (𝐵 ≺ (cf‘𝐴) → 𝐵 ∈ V)
57 breq1 4688 . . . . . . . . . . . 12 (𝑦 = 𝐵 → (𝑦 ≺ (cf‘𝐴) ↔ 𝐵 ≺ (cf‘𝐴)))
58 foeq2 6150 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ((𝑥𝐵 ↦ suc 𝑥):𝑦onto→ran (𝑥𝐵 ↦ suc 𝑥) ↔ (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥)))
59 breq2 4689 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝑦 ↔ ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝐵))
6058, 59imbi12d 333 . . . . . . . . . . . 12 (𝑦 = 𝐵 → (((𝑥𝐵 ↦ suc 𝑥):𝑦onto→ran (𝑥𝐵 ↦ suc 𝑥) → ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝑦) ↔ ((𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥) → ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝐵)))
6157, 60imbi12d 333 . . . . . . . . . . 11 (𝑦 = 𝐵 → ((𝑦 ≺ (cf‘𝐴) → ((𝑥𝐵 ↦ suc 𝑥):𝑦onto→ran (𝑥𝐵 ↦ suc 𝑥) → ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝑦)) ↔ (𝐵 ≺ (cf‘𝐴) → ((𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥) → ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝐵))))
62 cfon 9115 . . . . . . . . . . . . 13 (cf‘𝐴) ∈ On
63 sdomdom 8025 . . . . . . . . . . . . 13 (𝑦 ≺ (cf‘𝐴) → 𝑦 ≼ (cf‘𝐴))
64 ondomen 8898 . . . . . . . . . . . . 13 (((cf‘𝐴) ∈ On ∧ 𝑦 ≼ (cf‘𝐴)) → 𝑦 ∈ dom card)
6562, 63, 64sylancr 696 . . . . . . . . . . . 12 (𝑦 ≺ (cf‘𝐴) → 𝑦 ∈ dom card)
66 fodomnum 8918 . . . . . . . . . . . 12 (𝑦 ∈ dom card → ((𝑥𝐵 ↦ suc 𝑥):𝑦onto→ran (𝑥𝐵 ↦ suc 𝑥) → ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝑦))
6765, 66syl 17 . . . . . . . . . . 11 (𝑦 ≺ (cf‘𝐴) → ((𝑥𝐵 ↦ suc 𝑥):𝑦onto→ran (𝑥𝐵 ↦ suc 𝑥) → ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝑦))
6861, 67vtoclg 3297 . . . . . . . . . 10 (𝐵 ∈ V → (𝐵 ≺ (cf‘𝐴) → ((𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥) → ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝐵)))
6956, 68mpcom 38 . . . . . . . . 9 (𝐵 ≺ (cf‘𝐴) → ((𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥) → ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝐵))
7054, 69mpi 20 . . . . . . . 8 (𝐵 ≺ (cf‘𝐴) → ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝐵)
7151, 70syl5eqbrr 4721 . . . . . . 7 (𝐵 ≺ (cf‘𝐴) → {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ≼ 𝐵)
72 domtr 8050 . . . . . . 7 (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ≼ 𝐵) → (cf‘𝐴) ≼ 𝐵)
7371, 72sylan2 490 . . . . . 6 (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ∧ 𝐵 ≺ (cf‘𝐴)) → (cf‘𝐴) ≼ 𝐵)
74 domnsym 8127 . . . . . 6 ((cf‘𝐴) ≼ 𝐵 → ¬ 𝐵 ≺ (cf‘𝐴))
7573, 74syl 17 . . . . 5 (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ∧ 𝐵 ≺ (cf‘𝐴)) → ¬ 𝐵 ≺ (cf‘𝐴))
7675pm2.01da 457 . . . 4 ((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} → ¬ 𝐵 ≺ (cf‘𝐴))
7776a1i 11 . . 3 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} → ¬ 𝐵 ≺ (cf‘𝐴)))
7829, 49, 773syld 60 . 2 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ( 𝐵 = 𝐴 → ¬ 𝐵 ≺ (cf‘𝐴)))
7978necon2ad 2838 1 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  Vcvv 3231  wss 3607   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762  dom cdm 5143  ran crn 5144  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763   Fn wfn 5921  ontowfo 5924  cfv 5926  cdom 7995  csdm 7996  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-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-card 8803  df-cf 8805  df-acn 8806
This theorem is referenced by:  tskuni  9643
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