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Theorem prssg 4318
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 4296 . . 3 (𝐴𝑉 → (𝐴𝐶 ↔ {𝐴} ⊆ 𝐶))
2 snssg 4296 . . 3 (𝐵𝑊 → (𝐵𝐶 ↔ {𝐵} ⊆ 𝐶))
31, 2bi2anan9 916 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶)))
4 unss 3765 . . 3 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
5 df-pr 4151 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65sseq1i 3608 . . 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 384  wcel 1987  cun 3553  wss 3555  {csn 4148  {cpr 4150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3560  df-in 3562  df-ss 3569  df-sn 4149  df-pr 4151
This theorem is referenced by:  prss  4319  prssi  4321  prsspwg  4323  ssprss  4324  prelpw  4875  hashdmpropge2  13203  lspprss  18911  lspvadd  19015  topgele  20649  umgredgprv  25897  usgredgprvALT  25980  dfnbgr2  26122  nbuhgr  26126  uhgrnbgr0nb  26137  2wlkdlem6  26696  1wlkdlem2  26864  dihmeetlem2N  36065  fourierdlem20  39648  fourierdlem50  39677  fourierdlem54  39681  fourierdlem64  39691  fourierdlem76  39703  omeunle  40034
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