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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 2818 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | topopn 21442 | . . 3 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
3 | nordeq 6203 | . . . 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 33678 | . . 3 ⊢ (∪ 𝐽 ∈ On → suc ∪ 𝐽 ∈ Top) | |
7 | 6 | ordtoplem 33680 | . 2 ⊢ (Ord 𝐽 → (𝐽 ≠ ∪ 𝐽 → 𝐽 ∈ Top)) |
8 | 5, 7 | impbid 213 | 1 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 ≠ wne 3013 ∪ cuni 4830 Ord word 6183 Topctop 21429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fv 6356 df-topgen 16705 df-top 21430 df-bases 21482 |
This theorem is referenced by: ordtopconn 33684 ordtopt0 33687 ordcmp 33692 |
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