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Mirrors > Home > ILE Home > Th. List > ifssun | GIF version |
Description: A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.) |
Ref | Expression |
---|---|
ifssun | ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfif6 3518 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | |
2 | ssrab2 3223 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
3 | ssrab2 3223 | . . 3 ⊢ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} ⊆ 𝐵 | |
4 | unss12 3290 | . . 3 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} ⊆ 𝐵) → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴 ∪ 𝐵)) | |
5 | 2, 3, 4 | mp2an 423 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴 ∪ 𝐵) |
6 | 1, 5 | eqsstri 3170 | 1 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 {crab 2446 ∪ cun 3110 ⊆ wss 3112 ifcif 3516 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rab 2451 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-if 3517 |
This theorem is referenced by: ifidss 3531 ifelpwung 4454 |
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