Theorem prssg 4775

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 4740	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐶 ↔ {𝐴} ⊆ 𝐶))
2		snssg 4740	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ 𝐶 ↔ {𝐵} ⊆ 𝐶))
3	1, 2	bi2anan9 638	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶)))
4		unss 4142	. . 3 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
5		df-pr 4583	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
6	5	sseq1i 3962	. . 3 ⊢ ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
7	4, 6	bitr4i 278	. 2 ⊢ (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)
8	3, 7	bitrdi 287	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113   ∪ cun 3899   ⊆ wss 3901  {csn 4580  {cpr 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-un 3906  df-ss 3918  df-sn 4581  df-pr 4583

This theorem is referenced by:  prss  4776  prssi  4777  prsspwg  4779  ssprss  4780  prelpw  5394  hashdmpropge2  14406  lspvadd  21048  umgredgprv  29180  usgredgprvALT  29268  dfnbgr2  29410  nbuhgr  29416  uhgrnbgr0nb  29427  2wlkdlem6  30004  1wlkdlem2  30213  prssad  32604  prssbd  32605  tpssg  32612  coss0  38738  dihmeetlem2N  41555  mnuprdlem2  44510  fourierdlem20  46367  fourierdlem50  46396  fourierdlem54  46400  fourierdlem64  46410  fourierdlem76  46422  omeunle  46756  dfclnbgr2  48065  dfsclnbgr2  48088  dfvopnbgr2  48095  isubgr3stgrlem7  48214