Theorem ifssun 3620

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 3607	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
2		ssrab2 3312	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
3		ssrab2 3312	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} ⊆ 𝐵
4		unss12 3379	. . 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 3259	1 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:  ¬ wn 3  {crab 2514   ∪ cun 3198   ⊆ wss 3200  ifcif 3605

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-if 3606

This theorem is referenced by:  ifidss  3621  ifelpwung  4578