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Theorem prssg 4778
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 4743 . . 3 (𝐴𝑉 → (𝐴𝐶 ↔ {𝐴} ⊆ 𝐶))
2 snssg 4743 . . 3 (𝐵𝑊 → (𝐵𝐶 ↔ {𝐵} ⊆ 𝐶))
31, 2bi2anan9 638 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶)))
4 unss 4143 . . 3 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
5 df-pr 4588 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65sseq1i 3971 . . 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 205  wa 397  wcel 2107  cun 3907  wss 3909  {csn 4585  {cpr 4587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3446  df-un 3914  df-in 3916  df-ss 3926  df-sn 4586  df-pr 4588
This theorem is referenced by:  prss  4779  prssi  4780  prsspwg  4782  ssprss  4783  prelpw  5402  hashdmpropge2  14336  lspvadd  20510  umgredgprv  27887  usgredgprvALT  27972  dfnbgr2  28114  nbuhgr  28120  uhgrnbgr0nb  28131  2wlkdlem6  28705  1wlkdlem2  28911  coss0  36873  dihmeetlem2N  39694  mnuprdlem2  42458  fourierdlem20  44263  fourierdlem50  44292  fourierdlem54  44296  fourierdlem64  44306  fourierdlem76  44318  omeunle  44652
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