Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifcli | Structured version Visualization version GIF version |
Description: Membership (closure) of a conditional operator. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
ifcli.1 | ⊢ 𝐴 ∈ 𝐶 |
ifcli.2 | ⊢ 𝐵 ∈ 𝐶 |
Ref | Expression |
---|---|
ifcli | ⊢ if(𝜑, 𝐴, 𝐵) ∈ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifcli.1 | . 2 ⊢ 𝐴 ∈ 𝐶 | |
2 | ifcli.2 | . 2 ⊢ 𝐵 ∈ 𝐶 | |
3 | 1, 2 | keepel 4155 | 1 ⊢ if(𝜑, 𝐴, 𝐵) ∈ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 ifcif 4086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-if 4087 |
This theorem is referenced by: limsup10exlem 40004 |
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