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Theorem cfslb2n 9375
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 9373. 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 5997 . . . . . . . . . 10 (Lim 𝐴 → Ord 𝐴)
2 ordsson 7219 . . . . . . . . . 10 (Ord 𝐴𝐴 ⊆ On)
3 sstr 3806 . . . . . . . . . . 11 ((𝑥𝐴𝐴 ⊆ On) → 𝑥 ⊆ On)
43expcom 400 . . . . . . . . . 10 (𝐴 ⊆ On → (𝑥𝐴𝑥 ⊆ On))
51, 2, 43syl 18 . . . . . . . . 9 (Lim 𝐴 → (𝑥𝐴𝑥 ⊆ On))
6 onsucuni 7258 . . . . . . . . 9 (𝑥 ⊆ On → 𝑥 ⊆ suc 𝑥)
75, 6syl6 35 . . . . . . . 8 (Lim 𝐴 → (𝑥𝐴𝑥 ⊆ suc 𝑥))
87adantrd 481 . . . . . . 7 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥 ⊆ suc 𝑥))
98ralimdv 3151 . . . . . 6 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 𝑥 ⊆ suc 𝑥))
10 uniiun 4765 . . . . . . 7 𝐵 = 𝑥𝐵 𝑥
11 ss2iun 4728 . . . . . . 7 (∀𝑥𝐵 𝑥 ⊆ suc 𝑥 𝑥𝐵 𝑥 𝑥𝐵 suc 𝑥)
1210, 11syl5eqss 3846 . . . . . 6 (∀𝑥𝐵 𝑥 ⊆ suc 𝑥 𝐵 𝑥𝐵 suc 𝑥)
139, 12syl6 35 . . . . 5 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝐵 𝑥𝐵 suc 𝑥))
1413imp 395 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → 𝐵 𝑥𝐵 suc 𝑥)
15 cfslb.1 . . . . . . . . . 10 𝐴 ∈ V
1615cfslbn 9374 . . . . . . . . 9 ((Lim 𝐴𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐴)
17163expib 1145 . . . . . . . 8 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐴))
18 ordsucss 7248 . . . . . . . 8 (Ord 𝐴 → ( 𝑥𝐴 → suc 𝑥𝐴))
191, 17, 18sylsyld 61 . . . . . . 7 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → suc 𝑥𝐴))
2019ralimdv 3151 . . . . . 6 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 suc 𝑥𝐴))
21 iunss 4753 . . . . . 6 ( 𝑥𝐵 suc 𝑥𝐴 ↔ ∀𝑥𝐵 suc 𝑥𝐴)
2220, 21syl6ibr 243 . . . . 5 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐵 suc 𝑥𝐴))
2322imp 395 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → 𝑥𝐵 suc 𝑥𝐴)
24 sseq1 3823 . . . . . 6 ( 𝐵 = 𝐴 → ( 𝐵 𝑥𝐵 suc 𝑥𝐴 𝑥𝐵 suc 𝑥))
25 eqss 3813 . . . . . . 7 ( 𝑥𝐵 suc 𝑥 = 𝐴 ↔ ( 𝑥𝐵 suc 𝑥𝐴𝐴 𝑥𝐵 suc 𝑥))
2625simplbi2com 492 . . . . . 6 (𝐴 𝑥𝐵 suc 𝑥 → ( 𝑥𝐵 suc 𝑥𝐴 𝑥𝐵 suc 𝑥 = 𝐴))
2724, 26syl6bi 244 . . . . 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 7279 . . . . . . . . 9 (Lim 𝐴 → ( 𝑥𝐴 ↔ suc 𝑥𝐴))
3117, 30sylibd 230 . . . . . . . 8 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → suc 𝑥𝐴))
3231ralimdv 3151 . . . . . . 7 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 suc 𝑥𝐴))
3332imp 395 . . . . . 6 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ∀𝑥𝐵 suc 𝑥𝐴)
34 r19.29 3260 . . . . . . . 8 ((∀𝑥𝐵 suc 𝑥𝐴 ∧ ∃𝑥𝐵 𝑦 = suc 𝑥) → ∃𝑥𝐵 (suc 𝑥𝐴𝑦 = suc 𝑥))
35 eleq1 2873 . . . . . . . . . 10 (𝑦 = suc 𝑥 → (𝑦𝐴 ↔ suc 𝑥𝐴))
3635biimparc 467 . . . . . . . . 9 ((suc 𝑥𝐴𝑦 = suc 𝑥) → 𝑦𝐴)
3736rexlimivw 3217 . . . . . . . 8 (∃𝑥𝐵 (suc 𝑥𝐴𝑦 = suc 𝑥) → 𝑦𝐴)
3834, 37syl 17 . . . . . . 7 ((∀𝑥𝐵 suc 𝑥𝐴 ∧ ∃𝑥𝐵 𝑦 = suc 𝑥) → 𝑦𝐴)
3938ex 399 . . . . . 6 (∀𝑥𝐵 suc 𝑥𝐴 → (∃𝑥𝐵 𝑦 = suc 𝑥𝑦𝐴))
4033, 39syl 17 . . . . 5 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → (∃𝑥𝐵 𝑦 = suc 𝑥𝑦𝐴))
4140abssdv 3873 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴)
42 vuniex 7184 . . . . . . . 8 𝑥 ∈ V
4342sucex 7241 . . . . . . 7 suc 𝑥 ∈ V
4443dfiun2 4746 . . . . . 6 𝑥𝐵 suc 𝑥 = {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}
4544eqeq1i 2811 . . . . 5 ( 𝑥𝐵 suc 𝑥 = 𝐴 {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴)
4615cfslb 9373 . . . . . 6 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴 {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴) → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥})
47463expia 1143 . . . . 5 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴) → ( {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
4845, 47syl5bi 233 . . . 4 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴) → ( 𝑥𝐵 suc 𝑥 = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
4941, 48syldan 581 . . 3 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ( 𝑥𝐵 suc 𝑥 = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
50 eqid 2806 . . . . . . . . 9 (𝑥𝐵 ↦ suc 𝑥) = (𝑥𝐵 ↦ suc 𝑥)
5150rnmpt 5572 . . . . . . . 8 ran (𝑥𝐵 ↦ suc 𝑥) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}
5243, 50fnmpti 6233 . . . . . . . . . 10 (𝑥𝐵 ↦ suc 𝑥) Fn 𝐵
53 dffn4 6337 . . . . . . . . . 10 ((𝑥𝐵 ↦ suc 𝑥) Fn 𝐵 ↔ (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥))
5452, 53mpbi 221 . . . . . . . . 9 (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥)
55 relsdom 8199 . . . . . . . . . . 11 Rel ≺
5655brrelexi 5358 . . . . . . . . . 10 (𝐵 ≺ (cf‘𝐴) → 𝐵 ∈ V)
57 breq1 4847 . . . . . . . . . . . 12 (𝑦 = 𝐵 → (𝑦 ≺ (cf‘𝐴) ↔ 𝐵 ≺ (cf‘𝐴)))
58 foeq2 6328 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ((𝑥𝐵 ↦ suc 𝑥):𝑦onto→ran (𝑥𝐵 ↦ suc 𝑥) ↔ (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥)))
59 breq2 4848 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝑦 ↔ ran (𝑥𝐵 ↦ suc 𝑥) ≼ 𝐵))
6058, 59imbi12d 335 . . . . . . . . . . . 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 335 . . . . . . . . . . 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 9362 . . . . . . . . . . . . 13 (cf‘𝐴) ∈ On
63 sdomdom 8220 . . . . . . . . . . . . 13 (𝑦 ≺ (cf‘𝐴) → 𝑦 ≼ (cf‘𝐴))
64 ondomen 9143 . . . . . . . . . . . . 13 (((cf‘𝐴) ∈ On ∧ 𝑦 ≼ (cf‘𝐴)) → 𝑦 ∈ dom card)
6562, 63, 64sylancr 577 . . . . . . . . . . . 12 (𝑦 ≺ (cf‘𝐴) → 𝑦 ∈ dom card)
66 fodomnum 9163 . . . . . . . . . . . 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 3459 . . . . . . . . . 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 4880 . . . . . . 7 (𝐵 ≺ (cf‘𝐴) → {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ≼ 𝐵)
72 domtr 8245 . . . . . . 7 (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ≼ 𝐵) → (cf‘𝐴) ≼ 𝐵)
7371, 72sylan2 582 . . . . . 6 (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ∧ 𝐵 ≺ (cf‘𝐴)) → (cf‘𝐴) ≼ 𝐵)
74 domnsym 8325 . . . . . 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 824 . . . 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 2993 1 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1637  wcel 2156  {cab 2792  wne 2978  wral 3096  wrex 3097  Vcvv 3391  wss 3769   cuni 4630   ciun 4712   class class class wbr 4844  cmpt 4923  dom cdm 5311  ran crn 5312  Ord word 5935  Oncon0 5936  Lim wlim 5937  suc csuc 5938   Fn wfn 6096  ontowfo 6099  cfv 6101  cdom 8190  csdm 8191  cardccrd 9044  cfccf 9046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-er 7979  df-map 8094  df-en 8193  df-dom 8194  df-sdom 8195  df-card 9048  df-cf 9050  df-acn 9051
This theorem is referenced by:  tskuni  9890
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