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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsuctopon | Structured version Visualization version GIF version |
Description: One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.) |
Ref | Expression |
---|---|
onsuctopon | ⊢ (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsuctop 32432 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top) | |
2 | eloni 5733 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | ordunisuc 7032 | . . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
4 | 3 | eqcomd 2628 | . . 3 ⊢ (Ord 𝐴 → 𝐴 = ∪ suc 𝐴) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ On → 𝐴 = ∪ suc 𝐴) |
6 | istopon 20717 | . 2 ⊢ (suc 𝐴 ∈ (TopOn‘𝐴) ↔ (suc 𝐴 ∈ Top ∧ 𝐴 = ∪ suc 𝐴)) | |
7 | 1, 5, 6 | sylanbrc 698 | 1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∪ cuni 4436 Ord word 5722 Oncon0 5723 suc csuc 5725 ‘cfv 5888 Topctop 20698 TopOnctopon 20715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fv 5896 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 |
This theorem is referenced by: onsuct0 32440 |
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