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Theorem cfslb2n 9282
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 9280. 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 5945 . . . . . . . . . 10 (Lim 𝐴 → Ord 𝐴)
2 ordsson 7154 . . . . . . . . . 10 (Ord 𝐴𝐴 ⊆ On)
3 sstr 3752 . . . . . . . . . . 11 ((𝑥𝐴𝐴 ⊆ On) → 𝑥 ⊆ On)
43expcom 450 . . . . . . . . . 10 (𝐴 ⊆ On → (𝑥𝐴𝑥 ⊆ On))
51, 2, 43syl 18 . . . . . . . . 9 (Lim 𝐴 → (𝑥𝐴𝑥 ⊆ On))
6 onsucuni 7193 . . . . . . . . 9 (𝑥 ⊆ On → 𝑥 ⊆ suc 𝑥)
75, 6syl6 35 . . . . . . . 8 (Lim 𝐴 → (𝑥𝐴𝑥 ⊆ suc 𝑥))
87adantrd 485 . . . . . . 7 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥 ⊆ suc 𝑥))
98ralimdv 3101 . . . . . 6 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 𝑥 ⊆ suc 𝑥))
10 uniiun 4725 . . . . . . 7 𝐵 = 𝑥𝐵 𝑥
11 ss2iun 4688 . . . . . . 7 (∀𝑥𝐵 𝑥 ⊆ suc 𝑥 𝑥𝐵 𝑥 𝑥𝐵 suc 𝑥)
1210, 11syl5eqss 3790 . . . . . 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 9281 . . . . . . . . 9 ((Lim 𝐴𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐴)
17163expib 1117 . . . . . . . 8 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → 𝑥𝐴))
18 ordsucss 7183 . . . . . . . 8 (Ord 𝐴 → ( 𝑥𝐴 → suc 𝑥𝐴))
191, 17, 18sylsyld 61 . . . . . . 7 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → suc 𝑥𝐴))
2019ralimdv 3101 . . . . . 6 (Lim 𝐴 → (∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴)) → ∀𝑥𝐵 suc 𝑥𝐴))
21 iunss 4713 . . . . . 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 3767 . . . . . 6 ( 𝐵 = 𝐴 → ( 𝐵 𝑥𝐵 suc 𝑥𝐴 𝑥𝐵 suc 𝑥))
25 eqss 3759 . . . . . . 7 ( 𝑥𝐵 suc 𝑥 = 𝐴 ↔ ( 𝑥𝐵 suc 𝑥𝐴𝐴 𝑥𝐵 suc 𝑥))
2625simplbi2com 658 . . . . . 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 7214 . . . . . . . . 9 (Lim 𝐴 → ( 𝑥𝐴 ↔ suc 𝑥𝐴))
3117, 30sylibd 229 . . . . . . . 8 (Lim 𝐴 → ((𝑥𝐴𝑥 ≺ (cf‘𝐴)) → suc 𝑥𝐴))
3231ralimdv 3101 . . . . . . 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 3210 . . . . . . . 8 ((∀𝑥𝐵 suc 𝑥𝐴 ∧ ∃𝑥𝐵 𝑦 = suc 𝑥) → ∃𝑥𝐵 (suc 𝑥𝐴𝑦 = suc 𝑥))
35 eleq1 2827 . . . . . . . . . 10 (𝑦 = suc 𝑥 → (𝑦𝐴 ↔ suc 𝑥𝐴))
3635biimparc 505 . . . . . . . . 9 ((suc 𝑥𝐴𝑦 = suc 𝑥) → 𝑦𝐴)
3736rexlimivw 3167 . . . . . . . 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 3817 . . . 4 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴)
42 vuniex 7119 . . . . . . . 8 𝑥 ∈ V
4342sucex 7176 . . . . . . 7 suc 𝑥 ∈ V
4443dfiun2 4706 . . . . . 6 𝑥𝐵 suc 𝑥 = {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}
4544eqeq1i 2765 . . . . 5 ( 𝑥𝐵 suc 𝑥 = 𝐴 {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴)
4615cfslb 9280 . . . . . 6 ((Lim 𝐴 ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ⊆ 𝐴 {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} = 𝐴) → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥})
47463expia 1115 . . . . 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 488 . . 3 ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → ( 𝑥𝐵 suc 𝑥 = 𝐴 → (cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}))
50 eqid 2760 . . . . . . . . 9 (𝑥𝐵 ↦ suc 𝑥) = (𝑥𝐵 ↦ suc 𝑥)
5150rnmpt 5526 . . . . . . . 8 ran (𝑥𝐵 ↦ suc 𝑥) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥}
5243, 50fnmpti 6183 . . . . . . . . . 10 (𝑥𝐵 ↦ suc 𝑥) Fn 𝐵
53 dffn4 6282 . . . . . . . . . 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 8128 . . . . . . . . . . 11 Rel ≺
5655brrelexi 5315 . . . . . . . . . 10 (𝐵 ≺ (cf‘𝐴) → 𝐵 ∈ V)
57 breq1 4807 . . . . . . . . . . . 12 (𝑦 = 𝐵 → (𝑦 ≺ (cf‘𝐴) ↔ 𝐵 ≺ (cf‘𝐴)))
58 foeq2 6273 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ((𝑥𝐵 ↦ suc 𝑥):𝑦onto→ran (𝑥𝐵 ↦ suc 𝑥) ↔ (𝑥𝐵 ↦ suc 𝑥):𝐵onto→ran (𝑥𝐵 ↦ suc 𝑥)))
59 breq2 4808 . . . . . . . . . . . . 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 9269 . . . . . . . . . . . . 13 (cf‘𝐴) ∈ On
63 sdomdom 8149 . . . . . . . . . . . . 13 (𝑦 ≺ (cf‘𝐴) → 𝑦 ≼ (cf‘𝐴))
64 ondomen 9050 . . . . . . . . . . . . 13 (((cf‘𝐴) ∈ On ∧ 𝑦 ≼ (cf‘𝐴)) → 𝑦 ∈ dom card)
6562, 63, 64sylancr 698 . . . . . . . . . . . 12 (𝑦 ≺ (cf‘𝐴) → 𝑦 ∈ dom card)
66 fodomnum 9070 . . . . . . . . . . . 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 3406 . . . . . . . . . 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 4840 . . . . . . 7 (𝐵 ≺ (cf‘𝐴) → {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ≼ 𝐵)
72 domtr 8174 . . . . . . 7 (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ∧ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ≼ 𝐵) → (cf‘𝐴) ≼ 𝐵)
7371, 72sylan2 492 . . . . . 6 (((cf‘𝐴) ≼ {𝑦 ∣ ∃𝑥𝐵 𝑦 = suc 𝑥} ∧ 𝐵 ≺ (cf‘𝐴)) → (cf‘𝐴) ≼ 𝐵)
74 domnsym 8251 . . . . . 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 2947 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 1632  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  Vcvv 3340  wss 3715   cuni 4588   ciun 4672   class class class wbr 4804  cmpt 4881  dom cdm 5266  ran crn 5267  Ord word 5883  Oncon0 5884  Lim wlim 5885  suc csuc 5886   Fn wfn 6044  ontowfo 6047  cfv 6049  cdom 8119  csdm 8120  cardccrd 8951  cfccf 8953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-card 8955  df-cf 8957  df-acn 8958
This theorem is referenced by:  tskuni  9797
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