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Theorem limsucncmpi 32569
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 3243 . . . . 5 (suc 𝐴 ∈ Top → suc 𝐴 ∈ V)
2 sucexb 7051 . . . . 5 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 224 . . . 4 (suc 𝐴 ∈ Top → 𝐴 ∈ V)
4 sssucid 5840 . . . . 5 𝐴 ⊆ suc 𝐴
5 elpwg 4199 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 suc 𝐴𝐴 ⊆ suc 𝐴))
64, 5mpbiri 248 . . . 4 (𝐴 ∈ V → 𝐴 ∈ 𝒫 suc 𝐴)
7 limsucncmpi.1 . . . . . . 7 Lim 𝐴
8 limuni 5823 . . . . . . 7 (Lim 𝐴𝐴 = 𝐴)
97, 8ax-mp 5 . . . . . 6 𝐴 = 𝐴
10 elin 3829 . . . . . . . . . 10 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐴𝑧 ∈ Fin))
11 elpwi 4201 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
1211anim1i 591 . . . . . . . . . 10 ((𝑧 ∈ 𝒫 𝐴𝑧 ∈ Fin) → (𝑧𝐴𝑧 ∈ Fin))
1310, 12sylbi 207 . . . . . . . . 9 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → (𝑧𝐴𝑧 ∈ Fin))
14 nlim0 5821 . . . . . . . . . . . . . . . 16 ¬ Lim ∅
157, 142th 254 . . . . . . . . . . . . . . 15 (Lim 𝐴 ↔ ¬ Lim ∅)
16 xor3 371 . . . . . . . . . . . . . . 15 (¬ (Lim 𝐴 ↔ Lim ∅) ↔ (Lim 𝐴 ↔ ¬ Lim ∅))
1715, 16mpbir 221 . . . . . . . . . . . . . 14 ¬ (Lim 𝐴 ↔ Lim ∅)
18 limeq 5773 . . . . . . . . . . . . . . 15 (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
1918necon3bi 2849 . . . . . . . . . . . . . 14 (¬ (Lim 𝐴 ↔ Lim ∅) → 𝐴 ≠ ∅)
2017, 19ax-mp 5 . . . . . . . . . . . . 13 𝐴 ≠ ∅
21 uni0 4497 . . . . . . . . . . . . 13 ∅ = ∅
2220, 21neeqtrri 2896 . . . . . . . . . . . 12 𝐴
23 unieq 4476 . . . . . . . . . . . . 13 (𝑧 = ∅ → 𝑧 = ∅)
2423neeq2d 2883 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝐴 𝑧𝐴 ∅))
2522, 24mpbiri 248 . . . . . . . . . . 11 (𝑧 = ∅ → 𝐴 𝑧)
2625a1i 11 . . . . . . . . . 10 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 = ∅ → 𝐴 𝑧))
27 limord 5822 . . . . . . . . . . . . . 14 (Lim 𝐴 → Ord 𝐴)
28 ordsson 7031 . . . . . . . . . . . . . 14 (Ord 𝐴𝐴 ⊆ On)
297, 27, 28mp2b 10 . . . . . . . . . . . . 13 𝐴 ⊆ On
30 sstr2 3643 . . . . . . . . . . . . 13 (𝑧𝐴 → (𝐴 ⊆ On → 𝑧 ⊆ On))
3129, 30mpi 20 . . . . . . . . . . . 12 (𝑧𝐴𝑧 ⊆ On)
32 ordunifi 8251 . . . . . . . . . . . . 13 ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin ∧ 𝑧 ≠ ∅) → 𝑧𝑧)
33323expia 1286 . . . . . . . . . . . 12 ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝑧𝑧))
3431, 33sylan 487 . . . . . . . . . . 11 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝑧𝑧))
35 ssel 3630 . . . . . . . . . . . . 13 (𝑧𝐴 → ( 𝑧𝑧 𝑧𝐴))
367, 27ax-mp 5 . . . . . . . . . . . . . 14 Ord 𝐴
37 nordeq 5780 . . . . . . . . . . . . . 14 ((Ord 𝐴 𝑧𝐴) → 𝐴 𝑧)
3836, 37mpan 706 . . . . . . . . . . . . 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 2909 . . . . . . . . 9 ((𝑧𝐴𝑧 ∈ Fin) → 𝐴 𝑧)
4313, 42syl 17 . . . . . . . 8 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → 𝐴 𝑧)
4443neneqd 2828 . . . . . . 7 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → ¬ 𝐴 = 𝑧)
4544nrex 3029 . . . . . 6 ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧
46 unieq 4476 . . . . . . . . 9 (𝑦 = 𝐴 𝑦 = 𝐴)
4746eqeq2d 2661 . . . . . . . 8 (𝑦 = 𝐴 → (𝐴 = 𝑦𝐴 = 𝐴))
48 pweq 4194 . . . . . . . . . . 11 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4948ineq1d 3846 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝒫 𝑦 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
5049rexeqdv 3175 . . . . . . . . 9 (𝑦 = 𝐴 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧))
5150notbid 307 . . . . . . . 8 (𝑦 = 𝐴 → (¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧 ↔ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧))
5247, 51anbi12d 747 . . . . . . 7 (𝑦 = 𝐴 → ((𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧) ↔ (𝐴 = 𝐴 ∧ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧)))
5352rspcev 3340 . . . . . 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 721 . . . . 5 (𝐴 ∈ 𝒫 suc 𝐴 → ∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
55 rexanali 3027 . . . . 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 208 . . . 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 437 . . 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 220 . 2 ¬ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
60 ordunisuc 7074 . . . . 5 (Ord 𝐴 suc 𝐴 = 𝐴)
617, 27, 60mp2b 10 . . . 4 suc 𝐴 = 𝐴
6261eqcomi 2660 . . 3 𝐴 = suc 𝐴
6362iscmp 21239 . 2 (suc 𝐴 ∈ Comp ↔ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧)))
6459, 63mtbir 312 1 ¬ suc 𝐴 ∈ Comp
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  Fincfn 7997  Topctop 20746  Compccmp 21237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-fin 8001  df-cmp 21238
This theorem is referenced by:  limsucncmp  32570
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