Theorem snsspr1 3503

Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)

Assertion

Ref	Expression
snsspr1	⊢ {A} ⊆ {A, B}

Proof of Theorem snsspr1

Step	Hyp	Ref	Expression
1		ssun1 3100	. 2 ⊢ {A} ⊆ ({A} ∪ {B})
2		df-pr 3374	. 2 ⊢ {A, B} = ({A} ∪ {B})
3	1, 2	sseqtr4i 2972	1 ⊢ {A} ⊆ {A, B}

Syntax hints:   ∪ cun 2909   ⊆ wss 2911  {csn 3367  {cpr 3368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pr 3374

This theorem is referenced by:  snsstp1  3505  ssprr  3518  uniop  3983  op1stb  4175  op1stbg  4176  ltrelxr  6857