ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifssun GIF version

Theorem ifssun 3541
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 3529 . 2 if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
2 ssrab2 3233 . . 3 {𝑥𝐴𝜑} ⊆ 𝐴
3 ssrab2 3233 . . 3 {𝑥𝐵 ∣ ¬ 𝜑} ⊆ 𝐵
4 unss12 3300 . . 3 (({𝑥𝐴𝜑} ⊆ 𝐴 ∧ {𝑥𝐵 ∣ ¬ 𝜑} ⊆ 𝐵) → ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴𝐵))
52, 3, 4mp2an 424 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑}) ⊆ (𝐴𝐵)
61, 5eqsstri 3180 1 if(𝜑, 𝐴, 𝐵) ⊆ (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  {crab 2453  cun 3120  wss 3122  ifcif 3527
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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153
This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-rab 2458  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-if 3528
This theorem is referenced by:  ifidss  3542  ifelpwung  4467
  Copyright terms: Public domain W3C validator