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Theorem limsucncmpi 34371
Description: The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
Hypothesis
Ref Expression
limsucncmpi.1 Lim 𝐴
Assertion
Ref Expression
limsucncmpi ¬ suc 𝐴 ∈ Comp

Proof of Theorem limsucncmpi
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3426 . . . . 5 (suc 𝐴 ∈ Top → suc 𝐴 ∈ V)
2 sucexb 7588 . . . . 5 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 237 . . . 4 (suc 𝐴 ∈ Top → 𝐴 ∈ V)
4 sssucid 6290 . . . . 5 𝐴 ⊆ suc 𝐴
5 elpwg 4516 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 suc 𝐴𝐴 ⊆ suc 𝐴))
64, 5mpbiri 261 . . . 4 (𝐴 ∈ V → 𝐴 ∈ 𝒫 suc 𝐴)
7 limsucncmpi.1 . . . . . . 7 Lim 𝐴
8 limuni 6273 . . . . . . 7 (Lim 𝐴𝐴 = 𝐴)
97, 8ax-mp 5 . . . . . 6 𝐴 = 𝐴
10 elin 3882 . . . . . . . . . 10 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐴𝑧 ∈ Fin))
11 elpwi 4522 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
1211anim1i 618 . . . . . . . . . 10 ((𝑧 ∈ 𝒫 𝐴𝑧 ∈ Fin) → (𝑧𝐴𝑧 ∈ Fin))
1310, 12sylbi 220 . . . . . . . . 9 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → (𝑧𝐴𝑧 ∈ Fin))
14 nlim0 6271 . . . . . . . . . . . . . . . 16 ¬ Lim ∅
157, 142th 267 . . . . . . . . . . . . . . 15 (Lim 𝐴 ↔ ¬ Lim ∅)
16 xor3 387 . . . . . . . . . . . . . . 15 (¬ (Lim 𝐴 ↔ Lim ∅) ↔ (Lim 𝐴 ↔ ¬ Lim ∅))
1715, 16mpbir 234 . . . . . . . . . . . . . 14 ¬ (Lim 𝐴 ↔ Lim ∅)
18 limeq 6225 . . . . . . . . . . . . . . 15 (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
1918necon3bi 2967 . . . . . . . . . . . . . 14 (¬ (Lim 𝐴 ↔ Lim ∅) → 𝐴 ≠ ∅)
2017, 19ax-mp 5 . . . . . . . . . . . . 13 𝐴 ≠ ∅
21 uni0 4849 . . . . . . . . . . . . 13 ∅ = ∅
2220, 21neeqtrri 3014 . . . . . . . . . . . 12 𝐴
23 unieq 4830 . . . . . . . . . . . . 13 (𝑧 = ∅ → 𝑧 = ∅)
2423neeq2d 3001 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝐴 𝑧𝐴 ∅))
2522, 24mpbiri 261 . . . . . . . . . . 11 (𝑧 = ∅ → 𝐴 𝑧)
2625a1i 11 . . . . . . . . . 10 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 = ∅ → 𝐴 𝑧))
27 limord 6272 . . . . . . . . . . . . . 14 (Lim 𝐴 → Ord 𝐴)
28 ordsson 7567 . . . . . . . . . . . . . 14 (Ord 𝐴𝐴 ⊆ On)
297, 27, 28mp2b 10 . . . . . . . . . . . . 13 𝐴 ⊆ On
30 sstr2 3908 . . . . . . . . . . . . 13 (𝑧𝐴 → (𝐴 ⊆ On → 𝑧 ⊆ On))
3129, 30mpi 20 . . . . . . . . . . . 12 (𝑧𝐴𝑧 ⊆ On)
32 ordunifi 8921 . . . . . . . . . . . . 13 ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin ∧ 𝑧 ≠ ∅) → 𝑧𝑧)
33323expia 1123 . . . . . . . . . . . 12 ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝑧𝑧))
3431, 33sylan 583 . . . . . . . . . . 11 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝑧𝑧))
35 ssel 3893 . . . . . . . . . . . . 13 (𝑧𝐴 → ( 𝑧𝑧 𝑧𝐴))
367, 27ax-mp 5 . . . . . . . . . . . . . 14 Ord 𝐴
37 nordeq 6232 . . . . . . . . . . . . . 14 ((Ord 𝐴 𝑧𝐴) → 𝐴 𝑧)
3836, 37mpan 690 . . . . . . . . . . . . 13 ( 𝑧𝐴𝐴 𝑧)
3935, 38syl6 35 . . . . . . . . . . . 12 (𝑧𝐴 → ( 𝑧𝑧𝐴 𝑧))
4039adantr 484 . . . . . . . . . . 11 ((𝑧𝐴𝑧 ∈ Fin) → ( 𝑧𝑧𝐴 𝑧))
4134, 40syld 47 . . . . . . . . . 10 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝐴 𝑧))
4226, 41pm2.61dne 3028 . . . . . . . . 9 ((𝑧𝐴𝑧 ∈ Fin) → 𝐴 𝑧)
4313, 42syl 17 . . . . . . . 8 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → 𝐴 𝑧)
4443neneqd 2945 . . . . . . 7 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → ¬ 𝐴 = 𝑧)
4544nrex 3188 . . . . . 6 ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧
46 unieq 4830 . . . . . . . . 9 (𝑦 = 𝐴 𝑦 = 𝐴)
4746eqeq2d 2748 . . . . . . . 8 (𝑦 = 𝐴 → (𝐴 = 𝑦𝐴 = 𝐴))
48 pweq 4529 . . . . . . . . . . 11 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4948ineq1d 4126 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝒫 𝑦 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
5049rexeqdv 3326 . . . . . . . . 9 (𝑦 = 𝐴 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧))
5150notbid 321 . . . . . . . 8 (𝑦 = 𝐴 → (¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧 ↔ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧))
5247, 51anbi12d 634 . . . . . . 7 (𝑦 = 𝐴 → ((𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧) ↔ (𝐴 = 𝐴 ∧ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧)))
5352rspcev 3537 . . . . . 6 ((𝐴 ∈ 𝒫 suc 𝐴 ∧ (𝐴 = 𝐴 ∧ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧)) → ∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
549, 45, 53mpanr12 705 . . . . 5 (𝐴 ∈ 𝒫 suc 𝐴 → ∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
55 rexanali 3184 . . . . 5 (∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧) ↔ ¬ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
5654, 55sylib 221 . . . 4 (𝐴 ∈ 𝒫 suc 𝐴 → ¬ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
573, 6, 563syl 18 . . 3 (suc 𝐴 ∈ Top → ¬ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
58 imnan 403 . . 3 ((suc 𝐴 ∈ Top → ¬ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧)) ↔ ¬ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧)))
5957, 58mpbi 233 . 2 ¬ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
60 ordunisuc 7611 . . . . 5 (Ord 𝐴 suc 𝐴 = 𝐴)
617, 27, 60mp2b 10 . . . 4 suc 𝐴 = 𝐴
6261eqcomi 2746 . . 3 𝐴 = suc 𝐴
6362iscmp 22285 . 2 (suc 𝐴 ∈ Comp ↔ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧)))
6459, 63mtbir 326 1 ¬ suc 𝐴 ∈ Comp
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wne 2940  wral 3061  wrex 3062  Vcvv 3408  cin 3865  wss 3866  c0 4237  𝒫 cpw 4513   cuni 4819  Ord word 6212  Oncon0 6213  Lim wlim 6214  suc csuc 6215  Fincfn 8626  Topctop 21790  Compccmp 22283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-en 8627  df-fin 8630  df-cmp 22284
This theorem is referenced by:  limsucncmp  34372
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