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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 6414 | . . . 4 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → suc 𝐴 = suc if(Lim 𝐴, 𝐴, On)) | |
| 2 | 1 | eleq1d 2847 | . . 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 6358 | . . . 4 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → (Lim 𝐴 ↔ Lim if(Lim 𝐴, 𝐴, On))) | |
| 5 | limeq 6358 | . . . 4 ⊢ (On = if(Lim 𝐴, 𝐴, On) → (Lim On ↔ Lim if(Lim 𝐴, 𝐴, On))) | |
| 6 | limon 7816 | . . . 4 ⊢ Lim On | |
| 7 | 4, 5, 6 | elimhyp 4546 | . . 3 ⊢ Lim if(Lim 𝐴, 𝐴, On) |
| 8 | 7 | limsucncmpi 36805 | . 2 ⊢ ¬ suc if(Lim 𝐴, 𝐴, On) ∈ Comp |
| 9 | 3, 8 | dedth 4539 | 1 ⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1560 ∈ wcel 2142 ifcif 4480 Oncon0 6346 Lim wlim 6347 suc csuc 6348 Compccmp 23446 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pr 5390 ax-un 7718 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-om 7847 df-en 8928 df-fin 8931 df-cmp 23447 |
| This theorem is referenced by: ordcmp 36807 |
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