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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 2735 | . . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜑} | |
2 | 1 | dfif4 4546 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ {𝑥 ∣ 𝜑})) ∩ (𝐵 ∪ {𝑥 ∣ 𝜑}))) |
3 | inss1 4245 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ {𝑥 ∣ 𝜑})) ∩ (𝐵 ∪ {𝑥 ∣ 𝜑}))) ⊆ (𝐴 ∪ 𝐵) | |
4 | 2, 3 | eqsstri 4030 | 1 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: {cab 2712 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ifcif 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 |
This theorem is referenced by: (None) |
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