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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 6403 | . . . 4 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → suc 𝐴 = suc if(Lim 𝐴, 𝐴, On)) | |
| 2 | 1 | eleq1d 2841 | . . 3 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → (suc 𝐴 ∈ Comp ↔ suc if(Lim 𝐴, 𝐴, On) ∈ Comp)) |
| 3 | 2 | notbid 320 | . 2 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → (¬ suc 𝐴 ∈ Comp ↔ ¬ suc if(Lim 𝐴, 𝐴, On) ∈ Comp)) |
| 4 | limeq 6347 | . . . 4 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → (Lim 𝐴 ↔ Lim if(Lim 𝐴, 𝐴, On))) | |
| 5 | limeq 6347 | . . . 4 ⊢ (On = if(Lim 𝐴, 𝐴, On) → (Lim On ↔ Lim if(Lim 𝐴, 𝐴, On))) | |
| 6 | limon 7805 | . . . 4 ⊢ Lim On | |
| 7 | 4, 5, 6 | elimhyp 4540 | . . 3 ⊢ Lim if(Lim 𝐴, 𝐴, On) |
| 8 | 7 | limsucncmpi 36753 | . 2 ⊢ ¬ suc if(Lim 𝐴, 𝐴, On) ∈ Comp |
| 9 | 3, 8 | dedth 4533 | 1 ⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1554 ∈ wcel 2136 ifcif 4474 Oncon0 6335 Lim wlim 6336 suc csuc 6337 Compccmp 23419 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-sep 5240 ax-nul 5250 ax-pr 5384 ax-un 7707 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-ral 3071 df-rex 3081 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5095 df-opab 5157 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-om 7836 df-en 8917 df-fin 8920 df-cmp 23420 |
| This theorem is referenced by: ordcmp 36755 |
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