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Mirrors > Home > ILE Home > Th. List > ifidss | GIF version |
Description: A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.) |
Ref | Expression |
---|---|
ifidss | ⊢ if(𝜑, 𝐴, 𝐴) ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifssun 3540 | . 2 ⊢ if(𝜑, 𝐴, 𝐴) ⊆ (𝐴 ∪ 𝐴) | |
2 | unidm 3270 | . 2 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
3 | 1, 2 | sseqtri 3181 | 1 ⊢ if(𝜑, 𝐴, 𝐴) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3119 ⊆ wss 3121 ifcif 3526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 |
This theorem is referenced by: (None) |
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