Users' Mathboxes Mathbox for Chen-Pang He < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  limsucncmpi Structured version   Visualization version   GIF version

Theorem limsucncmpi 35820
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 3485 . . . . 5 (suc 𝐴 ∈ Top → suc 𝐴 ∈ V)
2 sucexb 7785 . . . . 5 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 233 . . . 4 (suc 𝐴 ∈ Top → 𝐴 ∈ V)
4 sssucid 6434 . . . . 5 𝐴 ⊆ suc 𝐴
5 elpwg 4597 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 suc 𝐴𝐴 ⊆ suc 𝐴))
64, 5mpbiri 258 . . . 4 (𝐴 ∈ V → 𝐴 ∈ 𝒫 suc 𝐴)
7 limsucncmpi.1 . . . . . . 7 Lim 𝐴
8 limuni 6415 . . . . . . 7 (Lim 𝐴𝐴 = 𝐴)
97, 8ax-mp 5 . . . . . 6 𝐴 = 𝐴
10 elin 3956 . . . . . . . . . 10 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐴𝑧 ∈ Fin))
11 elpwi 4601 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
1211anim1i 614 . . . . . . . . . 10 ((𝑧 ∈ 𝒫 𝐴𝑧 ∈ Fin) → (𝑧𝐴𝑧 ∈ Fin))
1310, 12sylbi 216 . . . . . . . . 9 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → (𝑧𝐴𝑧 ∈ Fin))
14 nlim0 6413 . . . . . . . . . . . . . . . 16 ¬ Lim ∅
157, 142th 264 . . . . . . . . . . . . . . 15 (Lim 𝐴 ↔ ¬ Lim ∅)
16 xor3 382 . . . . . . . . . . . . . . 15 (¬ (Lim 𝐴 ↔ Lim ∅) ↔ (Lim 𝐴 ↔ ¬ Lim ∅))
1715, 16mpbir 230 . . . . . . . . . . . . . 14 ¬ (Lim 𝐴 ↔ Lim ∅)
18 limeq 6366 . . . . . . . . . . . . . . 15 (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
1918necon3bi 2959 . . . . . . . . . . . . . 14 (¬ (Lim 𝐴 ↔ Lim ∅) → 𝐴 ≠ ∅)
2017, 19ax-mp 5 . . . . . . . . . . . . 13 𝐴 ≠ ∅
21 uni0 4929 . . . . . . . . . . . . 13 ∅ = ∅
2220, 21neeqtrri 3006 . . . . . . . . . . . 12 𝐴
23 unieq 4910 . . . . . . . . . . . . 13 (𝑧 = ∅ → 𝑧 = ∅)
2423neeq2d 2993 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝐴 𝑧𝐴 ∅))
2522, 24mpbiri 258 . . . . . . . . . . 11 (𝑧 = ∅ → 𝐴 𝑧)
2625a1i 11 . . . . . . . . . 10 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 = ∅ → 𝐴 𝑧))
27 limord 6414 . . . . . . . . . . . . . 14 (Lim 𝐴 → Ord 𝐴)
28 ordsson 7763 . . . . . . . . . . . . . 14 (Ord 𝐴𝐴 ⊆ On)
297, 27, 28mp2b 10 . . . . . . . . . . . . 13 𝐴 ⊆ On
30 sstr2 3981 . . . . . . . . . . . . 13 (𝑧𝐴 → (𝐴 ⊆ On → 𝑧 ⊆ On))
3129, 30mpi 20 . . . . . . . . . . . 12 (𝑧𝐴𝑧 ⊆ On)
32 ordunifi 9289 . . . . . . . . . . . . 13 ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin ∧ 𝑧 ≠ ∅) → 𝑧𝑧)
33323expia 1118 . . . . . . . . . . . 12 ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝑧𝑧))
3431, 33sylan 579 . . . . . . . . . . 11 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝑧𝑧))
35 ssel 3967 . . . . . . . . . . . . 13 (𝑧𝐴 → ( 𝑧𝑧 𝑧𝐴))
367, 27ax-mp 5 . . . . . . . . . . . . . 14 Ord 𝐴
37 nordeq 6373 . . . . . . . . . . . . . 14 ((Ord 𝐴 𝑧𝐴) → 𝐴 𝑧)
3836, 37mpan 687 . . . . . . . . . . . . 13 ( 𝑧𝐴𝐴 𝑧)
3935, 38syl6 35 . . . . . . . . . . . 12 (𝑧𝐴 → ( 𝑧𝑧𝐴 𝑧))
4039adantr 480 . . . . . . . . . . 11 ((𝑧𝐴𝑧 ∈ Fin) → ( 𝑧𝑧𝐴 𝑧))
4134, 40syld 47 . . . . . . . . . 10 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝐴 𝑧))
4226, 41pm2.61dne 3020 . . . . . . . . 9 ((𝑧𝐴𝑧 ∈ Fin) → 𝐴 𝑧)
4313, 42syl 17 . . . . . . . 8 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → 𝐴 𝑧)
4443neneqd 2937 . . . . . . 7 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → ¬ 𝐴 = 𝑧)
4544nrex 3066 . . . . . 6 ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧
46 unieq 4910 . . . . . . . . 9 (𝑦 = 𝐴 𝑦 = 𝐴)
4746eqeq2d 2735 . . . . . . . 8 (𝑦 = 𝐴 → (𝐴 = 𝑦𝐴 = 𝐴))
48 pweq 4608 . . . . . . . . . . 11 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4948ineq1d 4203 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝒫 𝑦 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
5049rexeqdv 3318 . . . . . . . . 9 (𝑦 = 𝐴 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧))
5150notbid 318 . . . . . . . 8 (𝑦 = 𝐴 → (¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧 ↔ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧))
5247, 51anbi12d 630 . . . . . . 7 (𝑦 = 𝐴 → ((𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧) ↔ (𝐴 = 𝐴 ∧ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧)))
5352rspcev 3604 . . . . . 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 702 . . . . 5 (𝐴 ∈ 𝒫 suc 𝐴 → ∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
55 rexanali 3094 . . . . 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 217 . . . 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 399 . . 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 229 . 2 ¬ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
60 ordunisuc 7813 . . . . 5 (Ord 𝐴 suc 𝐴 = 𝐴)
617, 27, 60mp2b 10 . . . 4 suc 𝐴 = 𝐴
6261eqcomi 2733 . . 3 𝐴 = suc 𝐴
6362iscmp 23214 . 2 (suc 𝐴 ∈ Comp ↔ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧)))
6459, 63mtbir 323 1 ¬ suc 𝐴 ∈ Comp
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wne 2932  wral 3053  wrex 3062  Vcvv 3466  cin 3939  wss 3940  c0 4314  𝒫 cpw 4594   cuni 4899  Ord word 6353  Oncon0 6354  Lim wlim 6355  suc csuc 6356  Fincfn 8935  Topctop 22717  Compccmp 23212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-om 7849  df-en 8936  df-fin 8939  df-cmp 23213
This theorem is referenced by:  limsucncmp  35821
  Copyright terms: Public domain W3C validator