Theorem ifssun 4514

Description: A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)

Assertion

Ref	Expression
ifssun	⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)

Proof of Theorem ifssun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜑}
2	1	dfif4 4512	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ {𝑥 ∣ 𝜑})) ∩ (𝐵 ∪ {𝑥 ∣ 𝜑})))
3		inss1 4208	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ {𝑥 ∣ 𝜑})) ∩ (𝐵 ∪ {𝑥 ∣ 𝜑}))) ⊆ (𝐴 ∪ 𝐵)
4	2, 3	eqsstri 4001	1 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:  {cab 2708  Vcvv 3455   ∖ cdif 3919   ∪ cun 3920   ∩ cin 3921   ⊆ wss 3922  ifcif 4496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497

This theorem is referenced by: (None)