Theorem ssdf2 44378

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 3980	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2875   ⊆ wss 3941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-v 3468  df-in 3948  df-ss 3958

This theorem is referenced by:  supminfxr2  44724  fsupdm  46103  finfdm  46107