Theorem ifssun 3597

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		dfif6 3584	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
2		ssrab2 3289	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
3		ssrab2 3289	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} ⊆ 𝐵
4		unss12 3356	. . 3 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} ⊆ 𝐵) → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴 ∪ 𝐵))
5	2, 3, 4	mp2an 426	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴 ∪ 𝐵)
6	1, 5	eqsstri 3236	1 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:  ¬ wn 3  {crab 2492   ∪ cun 3175   ⊆ wss 3177  ifcif 3582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rab 2497  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-if 3583

This theorem is referenced by:  ifidss  3598  ifelpwung  4549