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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 21607 | . . 3 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
2 | fvex 6682 | . . . 4 ⊢ ( I ‘𝐴) ∈ V | |
3 | indistopon 21608 | . . . 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 2910 | . 2 ⊢ {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)) |
6 | 5 | toponunii 21523 | 1 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 {cpr 4568 ∪ cuni 4837 I cid 5458 ‘cfv 6354 TopOnctopon 21517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-top 21501 df-topon 21518 |
This theorem is referenced by: indiscld 21698 indisconn 22025 txindis 22241 |
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