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Theorem snssiALT 39479
Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4447. This theorem was automatically generated from snssiALTVD 39478 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 4301 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2 eleq1a 2798 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
31, 2syl5bi 232 . . 3 (𝐴𝐵 → (𝑥 ∈ {𝐴} → 𝑥𝐵))
43alrimiv 1968 . 2 (𝐴𝐵 → ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
5 dfss2 3697 . 2 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
64, 5sylibr 224 1 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1594   = wceq 1596  wcel 2103  wss 3680  {csn 4285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-v 3306  df-in 3687  df-ss 3694  df-sn 4286
This theorem is referenced by: (None)
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