Theorem ssdf2 45500

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 412	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
6	1, 2, 3, 5	ssrd 3940	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884   ⊆ wss 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-11 2163  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-clel 2812  df-nfc 2886  df-ss 3920

This theorem is referenced by:  supminfxr2  45827  pimiooltgt  47068  fsupdm  47200  finfdm  47204