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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsucconn | Structured version Visualization version GIF version |
Description: A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
Ref | Expression |
---|---|
onsucconn | ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suceq 6437 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → suc 𝐴 = suc if(𝐴 ∈ On, 𝐴, ∅)) | |
2 | 1 | eleq1d 2810 | . 2 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (suc 𝐴 ∈ Conn ↔ suc if(𝐴 ∈ On, 𝐴, ∅) ∈ Conn)) |
3 | 0elon 6425 | . . . 4 ⊢ ∅ ∈ On | |
4 | 3 | elimel 4599 | . . 3 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On |
5 | 4 | onsucconni 36040 | . 2 ⊢ suc if(𝐴 ∈ On, 𝐴, ∅) ∈ Conn |
6 | 2, 5 | dedth 4588 | 1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∅c0 4322 ifcif 4530 Oncon0 6371 suc csuc 6373 Conncconn 23359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-ord 6374 df-on 6375 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-fv 6557 df-topgen 17428 df-top 22840 df-bases 22893 df-cld 22967 df-conn 23360 |
This theorem is referenced by: ordtopconn 36042 |
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