Metamath Proof Explorer


Theorem snssiALTVD

Description: Virtual deduction proof of snssiALT . (Contributed by Alan Sare, 11-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion snssiALTVD ( 𝐴𝐵 → { 𝐴 } ⊆ 𝐵 )

Proof

Step Hyp Ref Expression
1 dfss2 ( { 𝐴 } ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ { 𝐴 } → 𝑥𝐵 ) )
2 idn1 (    𝐴𝐵    ▶    𝐴𝐵    )
3 idn2 (    𝐴𝐵    ,    𝑥 ∈ { 𝐴 }    ▶    𝑥 ∈ { 𝐴 }    )
4 velsn ( 𝑥 ∈ { 𝐴 } ↔ 𝑥 = 𝐴 )
5 3 4 e2bi (    𝐴𝐵    ,    𝑥 ∈ { 𝐴 }    ▶    𝑥 = 𝐴    )
6 eleq1a ( 𝐴𝐵 → ( 𝑥 = 𝐴𝑥𝐵 ) )
7 2 5 6 e12 (    𝐴𝐵    ,    𝑥 ∈ { 𝐴 }    ▶    𝑥𝐵    )
8 7 in2 (    𝐴𝐵    ▶    ( 𝑥 ∈ { 𝐴 } → 𝑥𝐵 )    )
9 8 gen11 (    𝐴𝐵    ▶   𝑥 ( 𝑥 ∈ { 𝐴 } → 𝑥𝐵 )    )
10 biimpr ( ( { 𝐴 } ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ { 𝐴 } → 𝑥𝐵 ) ) → ( ∀ 𝑥 ( 𝑥 ∈ { 𝐴 } → 𝑥𝐵 ) → { 𝐴 } ⊆ 𝐵 ) )
11 1 9 10 e01 (    𝐴𝐵    ▶    { 𝐴 } ⊆ 𝐵    )
12 11 in1 ( 𝐴𝐵 → { 𝐴 } ⊆ 𝐵 )