Theorem ifssun 3655

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 3640	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
2		ssrab2 3333	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
3		ssrab2 3333	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} ⊆ 𝐵
4		unss12 3401	. . 3 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} ⊆ 𝐵) → ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴 ∪ 𝐵))
5	2, 3, 4	mp2an 430	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴 ∪ 𝐵)
6	1, 5	eqsstri 3280	1 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:  ¬ wn 3  {crab 2532   ∪ cun 3218   ⊆ wss 3220  ifcif 3638

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-if 3639

This theorem is referenced by:  ifidss  3656  ifelpwung  4625