Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2734 | . . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜑} | |
2 | 1 | dfif4 4444 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ {𝑥 ∣ 𝜑})) ∩ (𝐵 ∪ {𝑥 ∣ 𝜑}))) |
3 | inss1 4133 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ {𝑥 ∣ 𝜑})) ∩ (𝐵 ∪ {𝑥 ∣ 𝜑}))) ⊆ (𝐴 ∪ 𝐵) | |
4 | 2, 3 | eqsstri 3925 | 1 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: {cab 2712 Vcvv 3401 ∖ cdif 3854 ∪ cun 3855 ∩ cin 3856 ⊆ wss 3857 ifcif 4429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |