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 ∈ V → (A ∈ {A, B} ↔ (A = A ∨ A = B)))
5	3, 4	mpbiri 169	. . 3 ⊢ (A ∈ V → A ∈ {A, B})
6		snssi 1851	. . 3 ⊢ (A ∈ {A, B} → {A} ⊆ {A, B})
7	5, 6	syl 12	. 2 ⊢ (A ∈ V → {A} ⊆ {A, B})
8		snprc 1838	. . . 4 ⊢ (¬ A ∈ V ↔ {A} = ∅)
9	8	biimp 133	. . 3 ⊢ (¬ A ∈ V → {A} = ∅)
10		0ss 1725	. . . 4 ⊢ ∅ ⊆ {A, B}
11	10	a1i 7	. . 3 ⊢ (¬ A ∈ V → ∅ ⊆ {A, B})
12	9, 11	eqsstrd 1534	. 2 ⊢ (¬ A ∈ V → {A} ⊆ {A, B})
13	7, 12	pm2.61i 110	1 ⊢ {A} ⊆ {A, B}

Syntax hints:  ¬ wn 1   ∨ wo 195   = wceq 1091   ∈ 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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