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Theorem snssiALT 41531
Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4704. This theorem was automatically generated from snssiALTVD 41530 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 4544 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2 eleq1a 2888 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
31, 2syl5bi 245 . . 3 (𝐴𝐵 → (𝑥 ∈ {𝐴} → 𝑥𝐵))
43alrimiv 1928 . 2 (𝐴𝐵 → ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
5 dfss2 3904 . 2 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
64, 5sylibr 237 1 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536   = wceq 1538  wcel 2112  wss 3884  {csn 4528
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901  df-sn 4529
This theorem is referenced by: (None)
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