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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 6429 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → suc 𝐴 = suc if(𝐴 ∈ On, 𝐴, ∅)) | |
2 | suceq 6429 | . . . 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 2813 | . 2 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (suc suc 𝐴 ∈ Comp ↔ suc suc if(𝐴 ∈ On, 𝐴, ∅) ∈ Comp)) |
5 | 0elon 6417 | . . . 4 ⊢ ∅ ∈ On | |
6 | 5 | elimel 4593 | . . 3 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On |
7 | 6 | onsucsuccmpi 35863 | . 2 ⊢ suc suc if(𝐴 ∈ On, 𝐴, ∅) ∈ Comp |
8 | 4, 7 | dedth 4582 | 1 ⊢ (𝐴 ∈ On → suc suc 𝐴 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∅c0 4318 ifcif 4524 Oncon0 6363 suc csuc 6365 Compccmp 23277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7865 df-1o 8480 df-en 8956 df-fin 8959 df-topgen 17416 df-top 22783 df-bases 22836 df-cmp 23278 |
This theorem is referenced by: ordcmp 35867 |
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