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Mirrors > Home > MPE Home > Th. List > Mathboxes > ordtop | Structured version Visualization version GIF version |
Description: An ordinal is a topology iff it is not its supremum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.) |
Ref | Expression |
---|---|
ordtop | ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2740 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | topopn 22066 | . . 3 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
3 | nordeq 6284 | . . . 4 ⊢ ((Ord 𝐽 ∧ ∪ 𝐽 ∈ 𝐽) → 𝐽 ≠ ∪ 𝐽) | |
4 | 3 | ex 413 | . . 3 ⊢ (Ord 𝐽 → (∪ 𝐽 ∈ 𝐽 → 𝐽 ≠ ∪ 𝐽)) |
5 | 2, 4 | syl5 34 | . 2 ⊢ (Ord 𝐽 → (𝐽 ∈ Top → 𝐽 ≠ ∪ 𝐽)) |
6 | onsuctop 34631 | . . 3 ⊢ (∪ 𝐽 ∈ On → suc ∪ 𝐽 ∈ Top) | |
7 | 6 | ordtoplem 34633 | . 2 ⊢ (Ord 𝐽 → (𝐽 ≠ ∪ 𝐽 → 𝐽 ∈ Top)) |
8 | 5, 7 | impbid 211 | 1 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2110 ≠ wne 2945 ∪ cuni 4845 Ord word 6264 Topctop 22053 |
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 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-ord 6268 df-on 6269 df-suc 6271 df-iota 6390 df-fun 6434 df-fv 6440 df-topgen 17165 df-top 22054 df-bases 22107 |
This theorem is referenced by: ordtopconn 34637 ordtopt0 34640 ordcmp 34645 |
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