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Theorem prss 3547
Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypotheses
Ref Expression
prss.1 𝐴 ∈ V
prss.2 𝐵 ∈ V
Assertion
Ref Expression
prss ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)

Proof of Theorem prss
StepHypRef Expression
1 unss 3144 . 2 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
2 prss.1 . . . 4 𝐴 ∈ V
32snss 3521 . . 3 (𝐴𝐶 ↔ {𝐴} ⊆ 𝐶)
4 prss.2 . . . 4 𝐵 ∈ V
54snss 3521 . . 3 (𝐵𝐶 ↔ {𝐵} ⊆ 𝐶)
63, 5anbi12i 441 . 2 ((𝐴𝐶𝐵𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶))
7 df-pr 3409 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
87sseq1i 2996 . 2 ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
91, 6, 83bitr4i 205 1 ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wcel 1409  Vcvv 2574  cun 2942  wss 2944  {csn 3402  {cpr 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409
This theorem is referenced by:  tpss  3556  prsspw  3563
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