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Theorem prssg 4819
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 ((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))

Proof of Theorem prssg
StepHypRef Expression
1 snssg 4783 . . 3 (𝐴𝑉 → (𝐴𝐶 ↔ {𝐴} ⊆ 𝐶))
2 snssg 4783 . . 3 (𝐵𝑊 → (𝐵𝐶 ↔ {𝐵} ⊆ 𝐶))
31, 2bi2anan9 638 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶)))
4 unss 4190 . . 3 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
5 df-pr 4629 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65sseq1i 4012 . . 3 ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
74, 6bitr4i 278 . 2 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)
83, 7bitrdi 287 1 ((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  cun 3949  wss 3951  {csn 4626  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629
This theorem is referenced by:  prss  4820  prssi  4821  prsspwg  4823  ssprss  4824  prelpw  5451  hashdmpropge2  14522  lspvadd  21095  umgredgprv  29124  usgredgprvALT  29212  dfnbgr2  29354  nbuhgr  29360  uhgrnbgr0nb  29371  2wlkdlem6  29951  1wlkdlem2  30157  coss0  38480  dihmeetlem2N  41301  mnuprdlem2  44292  fourierdlem20  46142  fourierdlem50  46171  fourierdlem54  46175  fourierdlem64  46185  fourierdlem76  46197  omeunle  46531  dfclnbgr2  47810  dfsclnbgr2  47832  dfvopnbgr2  47839  isubgr3stgrlem7  47939
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