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Theorem cfslb2n 9741
 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 9739. 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 6233 . . . . . . . . . 10 (Lim 𝐴 → Ord 𝐴)
2 ordsson 7509 . . . . . . . . . 10 (Ord 𝐴𝐴 ⊆ On)
3 sstr 3902 . . . . . . . . . . 11 ((𝑥𝐴𝐴 ⊆ On) → 𝑥 ⊆ On)
43expcom 417 . . . . . . . . . 10 (𝐴 ⊆ On → (𝑥𝐴𝑥 ⊆ On))
51, 2, 43syl 18 . . . . . . . . 9 (Lim 𝐴 → (𝑥𝐴𝑥 ⊆ On))
6 onsucuni 7548 . . . . . . . . 9 (𝑥 ⊆ On → 𝑥 ⊆ suc 𝑥)
75, 6syl6 35 . . . . . . . 8 (Lim 𝐴 → (𝑥𝐴𝑥 ⊆ suc 𝑥))
87adantrd 495 . . . . . . 7 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥 ⊆ suc 𝑥))
98ralimdv 3109 . . . . . 6 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 𝑥 ⊆ suc 𝑥))
10 uniiun 4950 . . . . . . 7 𝐵 = 𝑥𝐵 𝑥
11 ss2iun 4904 . . . . . . 7 (∀𝑥𝐵 𝑥 ⊆ suc 𝑥 𝑥𝐵 𝑥 𝑥𝐵 suc 𝑥)
1210, 11eqsstrid 3942 . . . . . 6 (∀𝑥𝐵 𝑥 ⊆ suc 𝑥 𝐵 𝑥𝐵 suc 𝑥)
139, 12syl6 35 . . . . 5 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝐵 𝑥𝐵 suc 𝑥))
1413imp 410 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → 𝐵 𝑥𝐵 suc 𝑥)
15 cfslb.1 . . . . . . . . . 10 𝐴 ∈ V
1615cfslbn 9740 . . . . . . . . 9 ((Lim 𝐴𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐴)
17163expib 1119 . . . . . . . 8 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐴))
18 ordsucss 7538 . . . . . . . 8 (Ord 𝐴 → ( 𝑥𝐴 → suc 𝑥𝐴))
191, 17, 18sylsyld 61 . . . . . . 7 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → suc 𝑥𝐴))
2019ralimdv 3109 . . . . . 6 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 suc 𝑥𝐴))
21 iunss 4937 . . . . . 6 ( 𝑥𝐵 suc 𝑥𝐴 ↔ ∀𝑥𝐵 suc 𝑥𝐴)
2220, 21syl6ibr 255 . . . . 5 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐵 suc 𝑥𝐴))
2322imp 410 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → 𝑥𝐵 suc 𝑥𝐴)
24 sseq1 3919 . . . . . 6 ( 𝐵 = 𝐴 → ( 𝐵 𝑥𝐵 suc 𝑥𝐴 𝑥𝐵 suc 𝑥))
25 eqss 3909 . . . . . . 7 ( 𝑥𝐵 suc 𝑥 = 𝐴 ↔ ( 𝑥𝐵 suc 𝑥𝐴𝐴 𝑥𝐵 suc 𝑥))
2625simplbi2com 506 . . . . . 6 (𝐴 𝑥𝐵 suc 𝑥 → ( 𝑥𝐵 suc 𝑥𝐴 𝑥𝐵 suc 𝑥 = 𝐴))
2724, 26syl6bi 256 . . . . 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 7569 . . . . . . . . 9 (Lim 𝐴 → ( 𝑥𝐴 ↔ suc 𝑥𝐴))
3117, 30sylibd 242 . . . . . . . 8 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → suc 𝑥𝐴))
3231ralimdv 3109 . . . . . . 7 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 suc 𝑥𝐴))
3332imp 410 . . . . . 6 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ∀𝑥𝐵 suc 𝑥𝐴)
34 r19.29 3181 . . . . . . . 8 ((∀𝑥𝐵 suc 𝑥𝐴 ∧ ∃𝑥𝐵 𝑦 = suc 𝑥) → ∃𝑥𝐵 (suc 𝑥𝐴𝑦 = suc 𝑥))
35 eleq1 2839 . . . . . . . . . 10 (𝑦 = suc 𝑥 → (𝑦𝐴 ↔ suc 𝑥𝐴))
3635biimparc 483 . . . . . . . . 9 ((suc 𝑥𝐴𝑦 = suc 𝑥) → 𝑦𝐴)
3736rexlimivw 3206 . . . . . . . 8 (∃𝑥𝐵 (suc 𝑥𝐴𝑦 = suc 𝑥) → 𝑦𝐴)
3834, 37syl 17 . . . . . . 7 ((∀𝑥𝐵 suc 𝑥𝐴 ∧ ∃𝑥𝐵 𝑦 = suc 𝑥) → 𝑦𝐴)
3938ex 416 . . . . . 6 (∀𝑥𝐵 suc 𝑥𝐴 → (∃𝑥𝐵 𝑦 = suc 𝑥𝑦𝐴))
4033, 39syl 17 . . . . 5 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → (∃𝑥𝐵 𝑦 = suc 𝑥𝑦𝐴))
4140abssdv 3975 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴)
42 vuniex 7469 . . . . . . . 8 𝑥 ∈ V
4342sucex 7531 . . . . . . 7 suc 𝑥 ∈ V
4443dfiun2 4925 . . . . . 6 𝑥𝐵 suc 𝑥 = {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}
4544eqeq1i 2763 . . . . 5 ( 𝑥𝐵 suc 𝑥 = 𝐴 {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴)
4615cfslb 9739 . . . . . 6 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴 {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴) → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥})
47463expia 1118 . . . . 5 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴) → ( {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
4845, 47syl5bi 245 . . . 4 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴) → ( 𝑥𝐵 suc 𝑥 = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
4941, 48syldan 594 . . 3 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ( 𝑥𝐵 suc 𝑥 = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
50 eqid 2758 . . . . . . . . 9 (𝑥𝐵 ↦ suc 𝑥) = (𝑥𝐵 ↦ suc 𝑥)
5150rnmpt 5801 . . . . . . . 8 ran (𝑥𝐵 ↦ suc 𝑥) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}
5243, 50fnmpti 6479 . . . . . . . . . 10 (𝑥𝐵 ↦ suc 𝑥) Fn 𝐵
53 dffn4 6587 . . . . . . . . . 10 ((𝑥𝐵 ↦ suc 𝑥) Fn 𝐵 ↔ (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥))
5452, 53mpbi 233 . . . . . . . . 9 (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥)
55 relsdom 8547 . . . . . . . . . . 11 Rel ≺
5655brrelex1i 5582 . . . . . . . . . 10 (𝐵 ≺ (cf‘𝐴) → 𝐵 ∈ V)
57 breq1 5039 . . . . . . . . . . . 12 (𝑦 = 𝐵 → (𝑦 ≺ (cf‘𝐴) ↔ 𝐵 ≺ (cf‘𝐴)))
58 foeq2 6578 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ((𝑥𝐵 ↦ suc 𝑥):𝑦onto→ran (𝑥𝐵 ↦ suc 𝑥) ↔ (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥)))
59 breq2 5040 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝑦 ↔ ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝐵))
6058, 59imbi12d 348 . . . . . . . . . . . 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 348 . . . . . . . . . . 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 9728 . . . . . . . . . . . . 13 (cf‘𝐴) ∈ On
63 sdomdom 8568 . . . . . . . . . . . . 13 (𝑦 ≺ (cf‘𝐴) → 𝑦 ≼ (cf‘𝐴))
64 ondomen 9510 . . . . . . . . . . . . 13 (((cf‘𝐴) ∈ On ∧ 𝑦 ≼ (cf‘𝐴)) → 𝑦 ∈ dom card)
6562, 63, 64sylancr 590 . . . . . . . . . . . 12 (𝑦 ≺ (cf‘𝐴) → 𝑦 ∈ dom card)
66 fodomnum 9530 . . . . . . . . . . . 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 3487 . . . . . . . . . 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, 70eqbrtrrid 5072 . . . . . . 7 (𝐵 ≺ (cf‘𝐴) → {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ≼ 𝐵)
72 domtr 8593 . . . . . . 7 (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ≼ 𝐵) → (cf‘𝐴) ≼ 𝐵)
7371, 72sylan2 595 . . . . . 6 (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ∧ 𝐵 ≺ (cf‘𝐴)) → (cf‘𝐴) ≼ 𝐵)
74 domnsym 8678 . . . . . 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 798 . . . 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 2966 1 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → 𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ⊆ wss 3860  ∪ cuni 4801  ∪ ciun 4886   class class class wbr 5036   ↦ cmpt 5116  dom cdm 5528  ran crn 5529  Ord word 6173  Oncon0 6174  Lim wlim 6175  suc csuc 6176   Fn wfn 6335  –onto→wfo 6338  ‘cfv 6340   ≼ cdom 8538   ≺ csdm 8539  cardccrd 9410  cfccf 9412 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-isom 6349  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-er 8305  df-map 8424  df-en 8541  df-dom 8542  df-sdom 8543  df-card 9414  df-cf 9416  df-acn 9417 This theorem is referenced by:  tskuni  10256
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