Theorem ssdf2 45719

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 416	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
6	1, 2, 3, 5	ssrd 3941	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399  Ⅎwnf 1803   ∈ wcel 2142  Ⅎwnfc 2909   ⊆ wss 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-11 2191  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-nf 1804  df-clel 2837  df-nfc 2911  df-ss 3921

This theorem is referenced by:  supminfxr2  46043  pimiooltgt  47284  sssmf  47312  fsupdm  47416  finfdm  47420