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

Theorem snssiALT 40714
 Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4650. This theorem was automatically generated from snssiALTVD 40713 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 4490 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2 eleq1a 2877 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
31, 2syl5bi 243 . . 3 (𝐴𝐵 → (𝑥 ∈ {𝐴} → 𝑥𝐵))
43alrimiv 1906 . 2 (𝐴𝐵 → ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
5 dfss2 3879 . 2 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
64, 5sylibr 235 1 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1520   = wceq 1522   ∈ wcel 2080   ⊆ wss 3861  {csn 4474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-ext 2768 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-v 3438  df-in 3868  df-ss 3876  df-sn 4475 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator