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 35633
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 3491 . . . . 5 (suc 𝐴 ∈ Top → suc 𝐴 ∈ V)
2 sucexb 7794 . . . . 5 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 233 . . . 4 (suc 𝐴 ∈ Top → 𝐴 ∈ V)
4 sssucid 6443 . . . . 5 𝐴 ⊆ suc 𝐴
5 elpwg 4604 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 suc 𝐴𝐴 ⊆ suc 𝐴))
64, 5mpbiri 257 . . . 4 (𝐴 ∈ V → 𝐴 ∈ 𝒫 suc 𝐴)
7 limsucncmpi.1 . . . . . . 7 Lim 𝐴
8 limuni 6424 . . . . . . 7 (Lim 𝐴𝐴 = 𝐴)
97, 8ax-mp 5 . . . . . 6 𝐴 = 𝐴
10 elin 3963 . . . . . . . . . 10 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐴𝑧 ∈ Fin))
11 elpwi 4608 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
1211anim1i 613 . . . . . . . . . 10 ((𝑧 ∈ 𝒫 𝐴𝑧 ∈ Fin) → (𝑧𝐴𝑧 ∈ Fin))
1310, 12sylbi 216 . . . . . . . . 9 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → (𝑧𝐴𝑧 ∈ Fin))
14 nlim0 6422 . . . . . . . . . . . . . . . 16 ¬ Lim ∅
157, 142th 263 . . . . . . . . . . . . . . 15 (Lim 𝐴 ↔ ¬ Lim ∅)
16 xor3 381 . . . . . . . . . . . . . . 15 (¬ (Lim 𝐴 ↔ Lim ∅) ↔ (Lim 𝐴 ↔ ¬ Lim ∅))
1715, 16mpbir 230 . . . . . . . . . . . . . 14 ¬ (Lim 𝐴 ↔ Lim ∅)
18 limeq 6375 . . . . . . . . . . . . . . 15 (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
1918necon3bi 2965 . . . . . . . . . . . . . 14 (¬ (Lim 𝐴 ↔ Lim ∅) → 𝐴 ≠ ∅)
2017, 19ax-mp 5 . . . . . . . . . . . . 13 𝐴 ≠ ∅
21 uni0 4938 . . . . . . . . . . . . 13 ∅ = ∅
2220, 21neeqtrri 3012 . . . . . . . . . . . 12 𝐴
23 unieq 4918 . . . . . . . . . . . . 13 (𝑧 = ∅ → 𝑧 = ∅)
2423neeq2d 2999 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝐴 𝑧𝐴 ∅))
2522, 24mpbiri 257 . . . . . . . . . . 11 (𝑧 = ∅ → 𝐴 𝑧)
2625a1i 11 . . . . . . . . . 10 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 = ∅ → 𝐴 𝑧))
27 limord 6423 . . . . . . . . . . . . . 14 (Lim 𝐴 → Ord 𝐴)
28 ordsson 7772 . . . . . . . . . . . . . 14 (Ord 𝐴𝐴 ⊆ On)
297, 27, 28mp2b 10 . . . . . . . . . . . . 13 𝐴 ⊆ On
30 sstr2 3988 . . . . . . . . . . . . 13 (𝑧𝐴 → (𝐴 ⊆ On → 𝑧 ⊆ On))
3129, 30mpi 20 . . . . . . . . . . . 12 (𝑧𝐴𝑧 ⊆ On)
32 ordunifi 9295 . . . . . . . . . . . . 13 ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin ∧ 𝑧 ≠ ∅) → 𝑧𝑧)
33323expia 1119 . . . . . . . . . . . 12 ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝑧𝑧))
3431, 33sylan 578 . . . . . . . . . . 11 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝑧𝑧))
35 ssel 3974 . . . . . . . . . . . . 13 (𝑧𝐴 → ( 𝑧𝑧 𝑧𝐴))
367, 27ax-mp 5 . . . . . . . . . . . . . 14 Ord 𝐴
37 nordeq 6382 . . . . . . . . . . . . . 14 ((Ord 𝐴 𝑧𝐴) → 𝐴 𝑧)
3836, 37mpan 686 . . . . . . . . . . . . 13 ( 𝑧𝐴𝐴 𝑧)
3935, 38syl6 35 . . . . . . . . . . . 12 (𝑧𝐴 → ( 𝑧𝑧𝐴 𝑧))
4039adantr 479 . . . . . . . . . . 11 ((𝑧𝐴𝑧 ∈ Fin) → ( 𝑧𝑧𝐴 𝑧))
4134, 40syld 47 . . . . . . . . . 10 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝐴 𝑧))
4226, 41pm2.61dne 3026 . . . . . . . . 9 ((𝑧𝐴𝑧 ∈ Fin) → 𝐴 𝑧)
4313, 42syl 17 . . . . . . . 8 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → 𝐴 𝑧)
4443neneqd 2943 . . . . . . 7 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → ¬ 𝐴 = 𝑧)
4544nrex 3072 . . . . . 6 ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧
46 unieq 4918 . . . . . . . . 9 (𝑦 = 𝐴 𝑦 = 𝐴)
4746eqeq2d 2741 . . . . . . . 8 (𝑦 = 𝐴 → (𝐴 = 𝑦𝐴 = 𝐴))
48 pweq 4615 . . . . . . . . . . 11 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4948ineq1d 4210 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝒫 𝑦 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
5049rexeqdv 3324 . . . . . . . . 9 (𝑦 = 𝐴 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧))
5150notbid 317 . . . . . . . 8 (𝑦 = 𝐴 → (¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧 ↔ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧))
5247, 51anbi12d 629 . . . . . . 7 (𝑦 = 𝐴 → ((𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧) ↔ (𝐴 = 𝐴 ∧ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧)))
5352rspcev 3611 . . . . . 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 701 . . . . 5 (𝐴 ∈ 𝒫 suc 𝐴 → ∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
55 rexanali 3100 . . . . 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 398 . . 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 7822 . . . . 5 (Ord 𝐴 suc 𝐴 = 𝐴)
617, 27, 60mp2b 10 . . . 4 suc 𝐴 = 𝐴
6261eqcomi 2739 . . 3 𝐴 = suc 𝐴
6362iscmp 23112 . 2 (suc 𝐴 ∈ Comp ↔ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧)))
6459, 63mtbir 322 1 ¬ suc 𝐴 ∈ Comp
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wne 2938  wral 3059  wrex 3068  Vcvv 3472  cin 3946  wss 3947  c0 4321  𝒫 cpw 4601   cuni 4907  Ord word 6362  Oncon0 6363  Lim wlim 6364  suc csuc 6365  Fincfn 8941  Topctop 22615  Compccmp 23110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-en 8942  df-fin 8945  df-cmp 23111
This theorem is referenced by:  limsucncmp  35634
  Copyright terms: Public domain W3C validator