Theorem snsspr 2461

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 1468	. . . . 5 ⊢ A = A
2	1	orci 270	. . . 4 ⊢ (A = A ⋁ A = B)
3		elprg 2413	. . . 4 ⊢ (A ∈ V → (A ∈ {A, B} ↔ (A = A ⋁ A = B)))
4	2, 3	mpbiri 194	. . 3 ⊢ (A ∈ V → A ∈ {A, B})
5		snssi 2457	. . 3 ⊢ (A ∈ {A, B} → {A} ⊆ {A, B})
6	4, 5	syl 10	. 2 ⊢ (A ∈ V → {A} ⊆ {A, B})
7		snprc 2433	. . . 4 ⊢ (¬ A ∈ V ↔ {A} = ∅)
8	7	biimp 151	. . 3 ⊢ (¬ A ∈ V → {A} = ∅)
9		0ss 2291	. . . 4 ⊢ ∅ ⊆ {A, B}
10	9	a1i 8	. . 3 ⊢ (¬ A ∈ V → ∅ ⊆ {A, B})
11	8, 10	eqsstrd 2085	. 2 ⊢ (¬ A ∈ V → {A} ⊆ {A, B})
12	6, 11	pm2.61i 126	1 ⊢ {A} ⊆ {A, B}

Syntax hints:  ¬ wn 2   ⋁ wo 222   = wceq 953   ∈ wcel 955  Vcvv 1802   ⊆ wss 2037  ∅c0 2270  {csn 2399  {cpr 2400

This theorem is referenced by:  sspr 2466  uniop 2797  op1stb 2903  rankop 4665

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-sn 2402  df-pr 2403

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