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Mirrors > Home > MPE Home > Th. List > indisuni | Structured version Visualization version GIF version |
Description: The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
indisuni | ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indislem 22994 | . . 3 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
2 | fvex 6914 | . . . 4 ⊢ ( I ‘𝐴) ∈ V | |
3 | indistopon 22995 | . . . 4 ⊢ (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)) |
5 | 1, 4 | eqeltrri 2823 | . 2 ⊢ {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)) |
6 | 5 | toponunii 22909 | 1 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 {cpr 4635 ∪ cuni 4913 I cid 5579 ‘cfv 6554 TopOnctopon 22903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-top 22887 df-topon 22904 |
This theorem is referenced by: indiscld 23086 indisconn 23413 txindis 23629 |
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