Theorem prssg 4780

Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
prssg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))

Proof of Theorem prssg

Step	Hyp	Ref	Expression
1		snssg 4744	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶))
2		snssg 4744	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ 𝐶 ↔ {𝐵} ⊆ 𝐶))
3	1, 2	bi2anan9 650	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶)))
4		unss 4136	. . 3 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
5		df-pr 4587	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
6	5	sseq1i 3959	. . 3 ⊢ ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
7	4, 6	bitr4i 281	. 2 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)
8	3, 7	bitrdi 290	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145   ∪ cun 3897   ⊆ wss 3899  {csn 4584  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904  df-ss 3916  df-sn 4585  df-pr 4587

This theorem is used by:  prss  4781  prssi  4782  prsspwg  4784  ssprss  4785  prelpw  5421  hashdmpropge2  14549  lspvadd  21281  umgredgprv  29565  usgredgprvALT  29656  dfnbgr2  29798  nbuhgr  29804  uhgrnbgr0nb  29815  2wlkdlem6  30400  1wlkdlem2  30609  prssad  33005  prssbd  33006  tpssg  33013  coss0  39318  dihmeetlem2N  42173  mnuprdlem2  45098  fourierdlem20  46956  fourierdlem50  46985  fourierdlem54  46989  fourierdlem64  46999  fourierdlem76  47011  omeunle  47345  dfclnbgr2  48740  dfsclnbgr2  48763  dfvopnbgr2  48770  isubgr3stgrlem7  48889