Theorem snsspr 2470

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 1475	. . . . 5 |- A = A
2	1	orci 270	. . . 4 |- (A = A \/ A = B)
3		elprg 2423	. . . 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 2466	. . 3 |- (A e. {A, B} -> {A} (_ {A, B})
6	4, 5	syl 10	. 2 |- (A e. V -> {A} (_ {A, B})
7		snprc 2443	. . . 4 |- (-. A e. V <-> {A} = (/))
8	7	biimp 151	. . 3 |- (-. A e. V -> {A} = (/))
9		0ss 2301	. . . 4 |- (/) (_ {A, B}
10	9	a1i 8	. . 3 |- (-. A e. V -> (/) (_ {A, B})
11	8, 10	eqsstrd 2095	. 2 |- (-. A e. V -> {A} (_ {A, B})
12	6, 11	pm2.61i 126	1 |- {A} (_ {A, B}

Syntax hints:  -. wn 2   \/ wo 222   = wceq 956   e. wcel 958  Vcvv 1811   (_ wss 2047  (/)c0 2280  {csn 2409  {cpr 2410

This theorem is referenced by:  sspr 2475  uniop 2808  op1stb 2913  rankop 4693

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413

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