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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 6256 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → suc 𝐴 = suc if(𝐴 ∈ On, 𝐴, ∅)) | |
2 | 1 | eleq1d 2897 | . 2 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (suc 𝐴 ∈ Conn ↔ suc if(𝐴 ∈ On, 𝐴, ∅) ∈ Conn)) |
3 | 0elon 6244 | . . . 4 ⊢ ∅ ∈ On | |
4 | 3 | elimel 4534 | . . 3 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On |
5 | 4 | onsucconni 33785 | . 2 ⊢ suc if(𝐴 ∈ On, 𝐴, ∅) ∈ Conn |
6 | 2, 5 | dedth 4523 | 1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Conn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∅c0 4291 ifcif 4467 Oncon0 6191 suc csuc 6193 Conncconn 22019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 df-topgen 16717 df-top 21502 df-bases 21554 df-cld 21627 df-conn 22020 |
This theorem is referenced by: ordtopconn 33787 |
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