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| Mirrors > Home > MPE Home > Th. List > ifssun | Structured version Visualization version 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 | eqid 2736 | . . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜑} | |
| 2 | 1 | dfif4 4540 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ {𝑥 ∣ 𝜑})) ∩ (𝐵 ∪ {𝑥 ∣ 𝜑}))) | 
| 3 | inss1 4236 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ {𝑥 ∣ 𝜑})) ∩ (𝐵 ∪ {𝑥 ∣ 𝜑}))) ⊆ (𝐴 ∪ 𝐵) | |
| 4 | 2, 3 | eqsstri 4029 | 1 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: {cab 2713 Vcvv 3479 ∖ cdif 3947 ∪ cun 3948 ∩ cin 3949 ⊆ wss 3950 ifcif 4524 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 | 
| This theorem is referenced by: (None) | 
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