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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 2823 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | topopn 21516 | . . 3 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
3 | nordeq 6212 | . . . 4 ⊢ ((Ord 𝐽 ∧ ∪ 𝐽 ∈ 𝐽) → 𝐽 ≠ ∪ 𝐽) | |
4 | 3 | ex 415 | . . 3 ⊢ (Ord 𝐽 → (∪ 𝐽 ∈ 𝐽 → 𝐽 ≠ ∪ 𝐽)) |
5 | 2, 4 | syl5 34 | . 2 ⊢ (Ord 𝐽 → (𝐽 ∈ Top → 𝐽 ≠ ∪ 𝐽)) |
6 | onsuctop 33783 | . . 3 ⊢ (∪ 𝐽 ∈ On → suc ∪ 𝐽 ∈ Top) | |
7 | 6 | ordtoplem 33785 | . 2 ⊢ (Ord 𝐽 → (𝐽 ≠ ∪ 𝐽 → 𝐽 ∈ Top)) |
8 | 5, 7 | impbid 214 | 1 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 ≠ wne 3018 ∪ cuni 4840 Ord word 6192 Topctop 21503 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-ord 6196 df-on 6197 df-suc 6199 df-iota 6316 df-fun 6359 df-fv 6365 df-topgen 16719 df-top 21504 df-bases 21556 |
This theorem is referenced by: ordtopconn 33789 ordtopt0 33792 ordcmp 33797 |
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