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Theorem snssiALT 43892
Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4811. This theorem was automatically generated from snssiALTVD 43891 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snssiALT (𝐴𝐵 → {𝐴} ⊆ 𝐵)

Proof of Theorem snssiALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 velsn 4644 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2 eleq1a 2827 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
31, 2biimtrid 241 . . 3 (𝐴𝐵 → (𝑥 ∈ {𝐴} → 𝑥𝐵))
43alrimiv 1929 . 2 (𝐴𝐵 → ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
5 dfss2 3968 . 2 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
64, 5sylibr 233 1 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  wcel 2105  wss 3948  {csn 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965  df-sn 4629
This theorem is referenced by: (None)
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