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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 3529 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | |
2 | ssrab2 3233 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
3 | ssrab2 3233 | . . 3 ⊢ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} ⊆ 𝐵 | |
4 | unss12 3300 | . . 3 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} ⊆ 𝐵) → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴 ∪ 𝐵)) | |
5 | 2, 3, 4 | mp2an 424 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴 ∪ 𝐵) |
6 | 1, 5 | eqsstri 3180 | 1 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 {crab 2453 ∪ cun 3120 ⊆ wss 3122 ifcif 3527 |
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 705 ax-5 1441 ax-7 1442 ax-gen 1443 ax-ie1 1487 ax-ie2 1488 ax-8 1498 ax-10 1499 ax-11 1500 ax-i12 1501 ax-bndl 1503 ax-4 1504 ax-17 1520 ax-i9 1524 ax-ial 1528 ax-i5r 1529 ax-ext 2153 |
This theorem depends on definitions: df-bi 116 df-tru 1352 df-nf 1455 df-sb 1757 df-clab 2158 df-cleq 2164 df-clel 2167 df-nfc 2302 df-rab 2458 df-v 2733 df-un 3126 df-in 3128 df-ss 3135 df-if 3528 |
This theorem is referenced by: ifidss 3542 ifelpwung 4467 |
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