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Theorem limsucncmpi 32083
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 3198 . . . . 5 (suc 𝐴 ∈ Top → suc 𝐴 ∈ V)
2 sucexb 6956 . . . . 5 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 224 . . . 4 (suc 𝐴 ∈ Top → 𝐴 ∈ V)
4 sssucid 5761 . . . . 5 𝐴 ⊆ suc 𝐴
5 elpwg 4138 . . . . 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 5744 . . . . . . 7 (Lim 𝐴𝐴 = 𝐴)
97, 8ax-mp 5 . . . . . 6 𝐴 = 𝐴
10 elin 3774 . . . . . . . . . 10 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐴𝑧 ∈ Fin))
11 elpwi 4140 . . . . . . . . . . 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 5742 . . . . . . . . . . . . . . . 16 ¬ Lim ∅
157, 142th 254 . . . . . . . . . . . . . . 15 (Lim 𝐴 ↔ ¬ Lim ∅)
16 xor3 372 . . . . . . . . . . . . . . 15 (¬ (Lim 𝐴 ↔ Lim ∅) ↔ (Lim 𝐴 ↔ ¬ Lim ∅))
1715, 16mpbir 221 . . . . . . . . . . . . . 14 ¬ (Lim 𝐴 ↔ Lim ∅)
18 limeq 5694 . . . . . . . . . . . . . . 15 (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
1918necon3bi 2816 . . . . . . . . . . . . . 14 (¬ (Lim 𝐴 ↔ Lim ∅) → 𝐴 ≠ ∅)
2017, 19ax-mp 5 . . . . . . . . . . . . 13 𝐴 ≠ ∅
21 uni0 4431 . . . . . . . . . . . . 13 ∅ = ∅
2220, 21neeqtrri 2863 . . . . . . . . . . . 12 𝐴
23 unieq 4410 . . . . . . . . . . . . 13 (𝑧 = ∅ → 𝑧 = ∅)
2423neeq2d 2850 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝐴 𝑧𝐴 ∅))
2522, 24mpbiri 248 . . . . . . . . . . 11 (𝑧 = ∅ → 𝐴 𝑧)
2625a1i 11 . . . . . . . . . 10 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 = ∅ → 𝐴 𝑧))
27 limord 5743 . . . . . . . . . . . . . 14 (Lim 𝐴 → Ord 𝐴)
28 ordsson 6936 . . . . . . . . . . . . . 14 (Ord 𝐴𝐴 ⊆ On)
297, 27, 28mp2b 10 . . . . . . . . . . . . 13 𝐴 ⊆ On
30 sstr2 3590 . . . . . . . . . . . . 13 (𝑧𝐴 → (𝐴 ⊆ On → 𝑧 ⊆ On))
3129, 30mpi 20 . . . . . . . . . . . 12 (𝑧𝐴𝑧 ⊆ On)
32 ordunifi 8154 . . . . . . . . . . . . 13 ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin ∧ 𝑧 ≠ ∅) → 𝑧𝑧)
33323expia 1264 . . . . . . . . . . . 12 ((𝑧 ⊆ On ∧ 𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝑧𝑧))
3431, 33sylan 488 . . . . . . . . . . 11 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝑧𝑧))
35 ssel 3577 . . . . . . . . . . . . 13 (𝑧𝐴 → ( 𝑧𝑧 𝑧𝐴))
367, 27ax-mp 5 . . . . . . . . . . . . . 14 Ord 𝐴
37 nordeq 5701 . . . . . . . . . . . . . 14 ((Ord 𝐴 𝑧𝐴) → 𝐴 𝑧)
3836, 37mpan 705 . . . . . . . . . . . . 13 ( 𝑧𝐴𝐴 𝑧)
3935, 38syl6 35 . . . . . . . . . . . 12 (𝑧𝐴 → ( 𝑧𝑧𝐴 𝑧))
4039adantr 481 . . . . . . . . . . 11 ((𝑧𝐴𝑧 ∈ Fin) → ( 𝑧𝑧𝐴 𝑧))
4134, 40syld 47 . . . . . . . . . 10 ((𝑧𝐴𝑧 ∈ Fin) → (𝑧 ≠ ∅ → 𝐴 𝑧))
4226, 41pm2.61dne 2876 . . . . . . . . 9 ((𝑧𝐴𝑧 ∈ Fin) → 𝐴 𝑧)
4313, 42syl 17 . . . . . . . 8 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → 𝐴 𝑧)
4443neneqd 2795 . . . . . . 7 (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → ¬ 𝐴 = 𝑧)
4544nrex 2994 . . . . . 6 ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧
46 unieq 4410 . . . . . . . . 9 (𝑦 = 𝐴 𝑦 = 𝐴)
4746eqeq2d 2631 . . . . . . . 8 (𝑦 = 𝐴 → (𝐴 = 𝑦𝐴 = 𝐴))
48 pweq 4133 . . . . . . . . . . 11 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4948ineq1d 3791 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝒫 𝑦 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
5049rexeqdv 3134 . . . . . . . . 9 (𝑦 = 𝐴 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧))
5150notbid 308 . . . . . . . 8 (𝑦 = 𝐴 → (¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧 ↔ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧))
5247, 51anbi12d 746 . . . . . . 7 (𝑦 = 𝐴 → ((𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧) ↔ (𝐴 = 𝐴 ∧ ¬ ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐴 = 𝑧)))
5352rspcev 3295 . . . . . 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 720 . . . . 5 (𝐴 ∈ 𝒫 suc 𝐴 → ∃𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧))
55 rexanali 2992 . . . . 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 438 . . 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 6979 . . . . 5 (Ord 𝐴 suc 𝐴 = 𝐴)
617, 27, 60mp2b 10 . . . 4 suc 𝐴 = 𝐴
6261eqcomi 2630 . . 3 𝐴 = suc 𝐴
6362iscmp 21101 . 2 (suc 𝐴 ∈ Comp ↔ (suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc 𝐴(𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝐴 = 𝑧)))
6459, 63mtbir 313 1 ¬ suc 𝐴 ∈ Comp
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402  Ord word 5681  Oncon0 5682  Lim wlim 5683  suc csuc 5684  Fincfn 7899  Topctop 20617  Compccmp 21099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-er 7687  df-en 7900  df-fin 7903  df-cmp 21100
This theorem is referenced by:  limsucncmp  32084
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