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Theorem cfslb2n 9678
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 9676. 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 6243 . . . . . . . . . 10 (Lim 𝐴 → Ord 𝐴)
2 ordsson 7493 . . . . . . . . . 10 (Ord 𝐴𝐴 ⊆ On)
3 sstr 3972 . . . . . . . . . . 11 ((𝑥𝐴𝐴 ⊆ On) → 𝑥 ⊆ On)
43expcom 414 . . . . . . . . . 10 (𝐴 ⊆ On → (𝑥𝐴𝑥 ⊆ On))
51, 2, 43syl 18 . . . . . . . . 9 (Lim 𝐴 → (𝑥𝐴𝑥 ⊆ On))
6 onsucuni 7532 . . . . . . . . 9 (𝑥 ⊆ On → 𝑥 ⊆ suc 𝑥)
75, 6syl6 35 . . . . . . . 8 (Lim 𝐴 → (𝑥𝐴𝑥 ⊆ suc 𝑥))
87adantrd 492 . . . . . . 7 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥 ⊆ suc 𝑥))
98ralimdv 3175 . . . . . 6 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 𝑥 ⊆ suc 𝑥))
10 uniiun 4973 . . . . . . 7 𝐵 = 𝑥𝐵 𝑥
11 ss2iun 4928 . . . . . . 7 (∀𝑥𝐵 𝑥 ⊆ suc 𝑥 𝑥𝐵 𝑥 𝑥𝐵 suc 𝑥)
1210, 11eqsstrid 4012 . . . . . 6 (∀𝑥𝐵 𝑥 ⊆ suc 𝑥 𝐵 𝑥𝐵 suc 𝑥)
139, 12syl6 35 . . . . 5 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝐵 𝑥𝐵 suc 𝑥))
1413imp 407 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → 𝐵 𝑥𝐵 suc 𝑥)
15 cfslb.1 . . . . . . . . . 10 𝐴 ∈ V
1615cfslbn 9677 . . . . . . . . 9 ((Lim 𝐴𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐴)
17163expib 1114 . . . . . . . 8 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐴))
18 ordsucss 7522 . . . . . . . 8 (Ord 𝐴 → ( 𝑥𝐴 → suc 𝑥𝐴))
191, 17, 18sylsyld 61 . . . . . . 7 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → suc 𝑥𝐴))
2019ralimdv 3175 . . . . . 6 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 suc 𝑥𝐴))
21 iunss 4960 . . . . . 6 ( 𝑥𝐵 suc 𝑥𝐴 ↔ ∀𝑥𝐵 suc 𝑥𝐴)
2220, 21syl6ibr 253 . . . . 5 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐵 suc 𝑥𝐴))
2322imp 407 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → 𝑥𝐵 suc 𝑥𝐴)
24 sseq1 3989 . . . . . 6 ( 𝐵 = 𝐴 → ( 𝐵 𝑥𝐵 suc 𝑥𝐴 𝑥𝐵 suc 𝑥))
25 eqss 3979 . . . . . . 7 ( 𝑥𝐵 suc 𝑥 = 𝐴 ↔ ( 𝑥𝐵 suc 𝑥𝐴𝐴 𝑥𝐵 suc 𝑥))
2625simplbi2com 503 . . . . . 6 (𝐴 𝑥𝐵 suc 𝑥 → ( 𝑥𝐵 suc 𝑥𝐴 𝑥𝐵 suc 𝑥 = 𝐴))
2724, 26syl6bi 254 . . . . 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 7553 . . . . . . . . 9 (Lim 𝐴 → ( 𝑥𝐴 ↔ suc 𝑥𝐴))
3117, 30sylibd 240 . . . . . . . 8 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → suc 𝑥𝐴))
3231ralimdv 3175 . . . . . . 7 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 suc 𝑥𝐴))
3332imp 407 . . . . . 6 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ∀𝑥𝐵 suc 𝑥𝐴)
34 r19.29 3251 . . . . . . . 8 ((∀𝑥𝐵 suc 𝑥𝐴 ∧ ∃𝑥𝐵 𝑦 = suc 𝑥) → ∃𝑥𝐵 (suc 𝑥𝐴𝑦 = suc 𝑥))
35 eleq1 2897 . . . . . . . . . 10 (𝑦 = suc 𝑥 → (𝑦𝐴 ↔ suc 𝑥𝐴))
3635biimparc 480 . . . . . . . . 9 ((suc 𝑥𝐴𝑦 = suc 𝑥) → 𝑦𝐴)
3736rexlimivw 3279 . . . . . . . 8 (∃𝑥𝐵 (suc 𝑥𝐴𝑦 = suc 𝑥) → 𝑦𝐴)
3834, 37syl 17 . . . . . . 7 ((∀𝑥𝐵 suc 𝑥𝐴 ∧ ∃𝑥𝐵 𝑦 = suc 𝑥) → 𝑦𝐴)
3938ex 413 . . . . . 6 (∀𝑥𝐵 suc 𝑥𝐴 → (∃𝑥𝐵 𝑦 = suc 𝑥𝑦𝐴))
4033, 39syl 17 . . . . 5 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → (∃𝑥𝐵 𝑦 = suc 𝑥𝑦𝐴))
4140abssdv 4042 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴)
42 vuniex 7455 . . . . . . . 8 𝑥 ∈ V
4342sucex 7515 . . . . . . 7 suc 𝑥 ∈ V
4443dfiun2 4949 . . . . . 6 𝑥𝐵 suc 𝑥 = {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}
4544eqeq1i 2823 . . . . 5 ( 𝑥𝐵 suc 𝑥 = 𝐴 {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴)
4615cfslb 9676 . . . . . 6 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴 {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴) → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥})
47463expia 1113 . . . . 5 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴) → ( {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
4845, 47syl5bi 243 . . . 4 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴) → ( 𝑥𝐵 suc 𝑥 = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
4941, 48syldan 591 . . 3 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ( 𝑥𝐵 suc 𝑥 = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
50 eqid 2818 . . . . . . . . 9 (𝑥𝐵 ↦ suc 𝑥) = (𝑥𝐵 ↦ suc 𝑥)
5150rnmpt 5820 . . . . . . . 8 ran (𝑥𝐵 ↦ suc 𝑥) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}
5243, 50fnmpti 6484 . . . . . . . . . 10 (𝑥𝐵 ↦ suc 𝑥) Fn 𝐵
53 dffn4 6589 . . . . . . . . . 10 ((𝑥𝐵 ↦ suc 𝑥) Fn 𝐵 ↔ (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥))
5452, 53mpbi 231 . . . . . . . . 9 (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥)
55 relsdom 8504 . . . . . . . . . . 11 Rel ≺
5655brrelex1i 5601 . . . . . . . . . 10 (𝐵 ≺ (cf‘𝐴) → 𝐵 ∈ V)
57 breq1 5060 . . . . . . . . . . . 12 (𝑦 = 𝐵 → (𝑦 ≺ (cf‘𝐴) ↔ 𝐵 ≺ (cf‘𝐴)))
58 foeq2 6580 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ((𝑥𝐵 ↦ suc 𝑥):𝑦onto→ran (𝑥𝐵 ↦ suc 𝑥) ↔ (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥)))
59 breq2 5061 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝑦 ↔ ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝐵))
6058, 59imbi12d 346 . . . . . . . . . . . 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 346 . . . . . . . . . . 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 9665 . . . . . . . . . . . . 13 (cf‘𝐴) ∈ On
63 sdomdom 8525 . . . . . . . . . . . . 13 (𝑦 ≺ (cf‘𝐴) → 𝑦 ≼ (cf‘𝐴))
64 ondomen 9451 . . . . . . . . . . . . 13 (((cf‘𝐴) ∈ On ∧ 𝑦 ≼ (cf‘𝐴)) → 𝑦 ∈ dom card)
6562, 63, 64sylancr 587 . . . . . . . . . . . 12 (𝑦 ≺ (cf‘𝐴) → 𝑦 ∈ dom card)
66 fodomnum 9471 . . . . . . . . . . . 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 3565 . . . . . . . . . 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 5093 . . . . . . 7 (𝐵 ≺ (cf‘𝐴) → {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ≼ 𝐵)
72 domtr 8550 . . . . . . 7 (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ≼ 𝐵) → (cf‘𝐴) ≼ 𝐵)
7371, 72sylan2 592 . . . . . 6 (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ∧ 𝐵 ≺ (cf‘𝐴)) → (cf‘𝐴) ≼ 𝐵)
74 domnsym 8631 . . . . . 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 795 . . . 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 3028 1 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wral 3135  wrex 3136  Vcvv 3492  wss 3933   cuni 4830   ciun 4910   class class class wbr 5057  cmpt 5137  dom cdm 5548  ran crn 5549  Ord word 6183  Oncon0 6184  Lim wlim 6185  suc csuc 6186   Fn wfn 6343  ontowfo 6346  cfv 6348  cdom 8495  csdm 8496  cardccrd 9352  cfccf 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-card 9356  df-cf 9358  df-acn 9359
This theorem is referenced by:  tskuni  10193
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