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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 3520 | . 2 ⊢ if(𝜑, 𝐴, 𝐴) ⊆ (𝐴 ∪ 𝐴) | |
2 | unidm 3251 | . 2 ⊢ (𝐴 ∪ 𝐴) = 𝐴 | |
3 | 1, 2 | sseqtri 3162 | 1 ⊢ if(𝜑, 𝐴, 𝐴) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3100 ⊆ wss 3102 ifcif 3506 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rab 2444 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-if 3507 |
This theorem is referenced by: (None) |
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