Theorem if0ss 3642

Description: A conditional class with the False alternative being sent to the empty class is included in the class corresponding to the True alternative. (Contributed by BJ, 5-May-2026.)

Assertion

Ref	Expression
if0ss	⊢ if(𝜑, 𝐴, ∅) ⊆ 𝐴

Proof of Theorem if0ss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		if0ab 3641	. 2 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}
2	1	ssrab3 3334	1 ⊢ if(𝜑, 𝐴, ∅) ⊆ 𝐴

Syntax hints:   ⊆ wss 3220  ∅c0 3520  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-in1 623  ax-in2 624  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-fal 1408  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-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639

This theorem is referenced by:  if0elpw  4293