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Theorem prssg 4487
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 4451 . . 3 (𝐴𝑉 → (𝐴𝐶 ↔ {𝐴} ⊆ 𝐶))
2 snssg 4451 . . 3 (𝐵𝑊 → (𝐵𝐶 ↔ {𝐵} ⊆ 𝐶))
31, 2bi2anan9 953 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶)))
4 unss 3922 . . 3 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
5 df-pr 4316 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65sseq1i 3762 . . 3 ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
74, 6bitr4i 267 . 2 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)
83, 7syl6bb 276 1 ((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2131  cun 3705  wss 3707  {csn 4313  {cpr 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-v 3334  df-un 3712  df-in 3714  df-ss 3721  df-sn 4314  df-pr 4316
This theorem is referenced by:  prss  4488  prssi  4490  prsspwg  4492  ssprss  4493  prelpw  5055  hashdmpropge2  13449  lspprss  19186  lspvadd  19290  topgele  20928  umgredgprv  26193  usgredgprvALT  26278  dfnbgr2  26421  nbuhgr  26430  uhgrnbgr0nb  26441  2wlkdlem6  27043  1wlkdlem2  27282  coss0  34544  dihmeetlem2N  37082  fourierdlem20  40839  fourierdlem50  40868  fourierdlem54  40872  fourierdlem64  40882  fourierdlem76  40894  omeunle  41228
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