Theorem snsspr 2440

Description: A singleton is a subset of an unordered pair containing its member.

Assertion

Ref	Expression
snsspr	|- {A} (_ {A, B}

Proof of Theorem snsspr

Step	Hyp	Ref	Expression
1		eqid 1452	. . . . 5 |- A = A
2	1	orci 270	. . . 4 |- (A = A \/ A = B)
3		elprg 2394	. . . 4 |- (A e. V -> (A e. {A, B} <-> (A = A \/ A = B)))
4	2, 3	mpbiri 194	. . 3 |- (A e. V -> A e. {A, B})
5		snssi 2436	. . 3 |- (A e. {A, B} -> {A} (_ {A, B})
6	4, 5	syl 10	. 2 |- (A e. V -> {A} (_ {A, B})
7		snprc 2414	. . . 4 |- (-. A e. V <-> {A} = (/))
8	7	biimp 151	. . 3 |- (-. A e. V -> {A} = (/))
9		0ss 2272	. . . 4 |- (/) (_ {A, B}
10	9	a1i 8	. . 3 |- (-. A e. V -> (/) (_ {A, B})
11	8, 10	eqsstrd 2066	. 2 |- (-. A e. V -> {A} (_ {A, B})
12	6, 11	pm2.61i 126	1 |- {A} (_ {A, B}

Syntax hints:  -. wn 2   \/ wo 222   = wceq 1099   e. wcel 1105  Vcvv 1786   (_ wss 2018  (/)c0 2251  {csn 2380  {cpr 2381

This theorem is referenced by:  sspr 2445  uniop 2771  op1stb 2876  rankop 4617

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-sn 2383  df-pr 2384

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