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| Mirrors > Home > MPE Home > Th. List > Mathboxes > limsucncmp | Structured version Visualization version GIF version | ||
| Description: The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.) |
| Ref | Expression |
|---|---|
| limsucncmp | ⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | suceq 6450 | . . . 4 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → suc 𝐴 = suc if(Lim 𝐴, 𝐴, On)) | |
| 2 | 1 | eleq1d 2826 | . . 3 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → (suc 𝐴 ∈ Comp ↔ suc if(Lim 𝐴, 𝐴, On) ∈ Comp)) |
| 3 | 2 | notbid 318 | . 2 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → (¬ suc 𝐴 ∈ Comp ↔ ¬ suc if(Lim 𝐴, 𝐴, On) ∈ Comp)) |
| 4 | limeq 6396 | . . . 4 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → (Lim 𝐴 ↔ Lim if(Lim 𝐴, 𝐴, On))) | |
| 5 | limeq 6396 | . . . 4 ⊢ (On = if(Lim 𝐴, 𝐴, On) → (Lim On ↔ Lim if(Lim 𝐴, 𝐴, On))) | |
| 6 | limon 7856 | . . . 4 ⊢ Lim On | |
| 7 | 4, 5, 6 | elimhyp 4591 | . . 3 ⊢ Lim if(Lim 𝐴, 𝐴, On) |
| 8 | 7 | limsucncmpi 36446 | . 2 ⊢ ¬ suc if(Lim 𝐴, 𝐴, On) ∈ Comp |
| 9 | 3, 8 | dedth 4584 | 1 ⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ifcif 4525 Oncon0 6384 Lim wlim 6385 suc csuc 6386 Compccmp 23394 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-en 8986 df-fin 8989 df-cmp 23395 |
| This theorem is referenced by: ordcmp 36448 |
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