Theorem snsspr 1853

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		cleqid 1102	. . . . . 6 |- A = A
2	1	pm2.21ni 92	. . . . 5 |- (-. A = A -> A = B)
3	2	orri 201	. . . 4 |- (A = A \/ A = B)
4		elprg 1822	. . . 4 |- (A e. V -> (A e. {A, B} <-> (A = A \/ A = B)))
5	3, 4	mpbiri 169	. . 3 |- (A e. V -> A e. {A, B})
6		snssi 1851	. . 3 |- (A e. {A, B} -> {A} (_ {A, B})
7	5, 6	syl 12	. 2 |- (A e. V -> {A} (_ {A, B})
8		snprc 1838	. . . 4 |- (-. A e. V <-> {A} = (/))
9	8	biimp 133	. . 3 |- (-. A e. V -> {A} = (/))
10		0ss 1725	. . . 4 |- (/) (_ {A, B}
11	10	a1i 7	. . 3 |- (-. A e. V -> (/) (_ {A, B})
12	9, 11	eqsstrd 1534	. 2 |- (-. A e. V -> {A} (_ {A, B})
13	7, 12	pm2.61i 110	1 |- {A} (_ {A, B}

Syntax hints:  -. wn 1   \/ wo 195   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  (/)c0 1707  {csn 1808  {cpr 1809

This theorem is referenced by:  unop 1931  op1stb 1992

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812

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