Theorem ifssun 4498

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 2762	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜑}
2	1	dfif4 4496	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ {𝑥 ∣ 𝜑})) ∩ (𝐵 ∪ {𝑥 ∣ 𝜑})))
3		inss1 4188	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ {𝑥 ∣ 𝜑})) ∩ (𝐵 ∪ {𝑥 ∣ 𝜑}))) ⊆ (𝐴 ∪ 𝐵)
4	2, 3	eqsstri 3982	1 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:  {cab 2740  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ifcif 4480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481

This theorem is referenced by: (None)