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Theorem prssg 3713
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 3692 . . 3 (𝐴𝑉 → (𝐴𝐶 ↔ {𝐴} ⊆ 𝐶))
2 snssg 3692 . . 3 (𝐵𝑊 → (𝐵𝐶 ↔ {𝐵} ⊆ 𝐶))
31, 2bi2anan9 596 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶)))
4 unss 3281 . . 3 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
5 df-pr 3567 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65sseq1i 3154 . . 3 ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
74, 6bitr4i 186 . 2 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)
83, 7bitrdi 195 1 ((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 2128  cun 3100  wss 3102  {csn 3560  {cpr 3561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567
This theorem is referenced by:  prssi  3714  prsspwg  3715  topgele  12438
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