Theorem prssg 4783

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 4747	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶))
2		snssg 4747	. . 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 4139	. . 3 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
5		df-pr 4590	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
6	5	sseq1i 3962	. . 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 3900   ⊆ wss 3902  {csn 4587  {cpr 4589

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 2734

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 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-ss 3919  df-sn 4588  df-pr 4590

This theorem is used by:  prss  4784  prssi  4785  prsspwg  4787  ssprss  4788  prelpw  5425  hashdmpropge2  14552  lspvadd  21286  umgredgprv  29572  usgredgprvALT  29663  dfnbgr2  29805  nbuhgr  29811  uhgrnbgr0nb  29822  2wlkdlem6  30407  1wlkdlem2  30616  prssad  33012  prssbd  33013  tpssg  33020  coss0  39325  dihmeetlem2N  42180  mnuprdlem2  45105  fourierdlem20  46963  fourierdlem50  46992  fourierdlem54  46996  fourierdlem64  47006  fourierdlem76  47018  omeunle  47352  dfclnbgr2  48747  dfsclnbgr2  48770  dfvopnbgr2  48777  isubgr3stgrlem7  48896