Theorem ssdf2 42690

Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
ssdf2.p	⊢ Ⅎ𝑥𝜑
ssdf2.a	⊢ Ⅎ𝑥𝐴
ssdf2.b	⊢ Ⅎ𝑥𝐵
ssdf2.x	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)

Assertion

Ref	Expression
ssdf2	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Proof of Theorem ssdf2

Step	Hyp	Ref	Expression
1		ssdf2.p	. 2 ⊢ Ⅎ𝑥𝜑
2		ssdf2.a	. 2 ⊢ Ⅎ𝑥𝐴
3		ssdf2.b	. 2 ⊢ Ⅎ𝑥𝐵
4		ssdf2.x	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
5	4	ex 413	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
6	1, 2, 3, 5	ssrd 3926	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887   ⊆ wss 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-v 3434  df-in 3894  df-ss 3904

This theorem is referenced by:  supminfxr2  43009