Theorem ifssun 3560

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 3548	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
2		ssrab2 3252	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
3		ssrab2 3252	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} ⊆ 𝐵
4		unss12 3319	. . 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 3199	1 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:  ¬ wn 3  {crab 2469   ∪ cun 3139   ⊆ wss 3141  ifcif 3546

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-if 3547

This theorem is referenced by:  ifidss  3561  ifelpwung  4493