Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  snssiALT Structured version   Visualization version   GIF version

Theorem snssiALT 44848
Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4808. This theorem was automatically generated from snssiALTVD 44847 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 4642 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2 eleq1a 2836 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
31, 2biimtrid 242 . . 3 (𝐴𝐵 → (𝑥 ∈ {𝐴} → 𝑥𝐵))
43alrimiv 1927 . 2 (𝐴𝐵 → ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
5 df-ss 3968 . 2 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
64, 5sylibr 234 1 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  wcel 2108  wss 3951  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-sn 4627
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator