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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsucsuccmp | Structured version Visualization version GIF version |
Description: The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.) |
Ref | Expression |
---|---|
onsucsuccmp | ⊢ (𝐴 ∈ On → suc suc 𝐴 ∈ Comp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suceq 6231 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → suc 𝐴 = suc if(𝐴 ∈ On, 𝐴, ∅)) | |
2 | suceq 6231 | . . . 4 ⊢ (suc 𝐴 = suc if(𝐴 ∈ On, 𝐴, ∅) → suc suc 𝐴 = suc suc if(𝐴 ∈ On, 𝐴, ∅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → suc suc 𝐴 = suc suc if(𝐴 ∈ On, 𝐴, ∅)) |
4 | 3 | eleq1d 2817 | . 2 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (suc suc 𝐴 ∈ Comp ↔ suc suc if(𝐴 ∈ On, 𝐴, ∅) ∈ Comp)) |
5 | 0elon 6219 | . . . 4 ⊢ ∅ ∈ On | |
6 | 5 | elimel 4480 | . . 3 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On |
7 | 6 | onsucsuccmpi 34262 | . 2 ⊢ suc suc if(𝐴 ∈ On, 𝐴, ∅) ∈ Comp |
8 | 4, 7 | dedth 4469 | 1 ⊢ (𝐴 ∈ On → suc suc 𝐴 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ∅c0 4209 ifcif 4411 Oncon0 6166 suc csuc 6168 Compccmp 22130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-om 7594 df-1o 8124 df-en 8549 df-fin 8552 df-topgen 16813 df-top 21638 df-bases 21690 df-cmp 22131 |
This theorem is referenced by: ordcmp 34266 |
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