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Theorem ifssun 3530
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) ⊆ (𝐴𝐵)

Proof of Theorem ifssun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfif6 3518 . 2 if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
2 ssrab2 3223 . . 3 {𝑥𝐴𝜑} ⊆ 𝐴
3 ssrab2 3223 . . 3 {𝑥𝐵 ∣ ¬ 𝜑} ⊆ 𝐵
4 unss12 3290 . . 3 (({𝑥𝐴𝜑} ⊆ 𝐴 ∧ {𝑥𝐵 ∣ ¬ 𝜑} ⊆ 𝐵) → ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴𝐵))
52, 3, 4mp2an 423 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴𝐵)
61, 5eqsstri 3170 1 if(𝜑, 𝐴, 𝐵) ⊆ (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  {crab 2446  cun 3110  wss 3112  ifcif 3516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rab 2451  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-if 3517
This theorem is referenced by:  ifidss  3531  ifelpwung  4454
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