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Theorem prss 3683
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 3254 . 2 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
2 prss.1 . . . 4 𝐴 ∈ V
32snss 3656 . . 3 (𝐴𝐶 ↔ {𝐴} ⊆ 𝐶)
4 prss.2 . . . 4 𝐵 ∈ V
54snss 3656 . . 3 (𝐵𝐶 ↔ {𝐵} ⊆ 𝐶)
63, 5anbi12i 456 . 2 ((𝐴𝐶𝐵𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶))
7 df-pr 3538 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
87sseq1i 3127 . 2 ({𝐴, 𝐵} ⊆ 𝐶 ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
91, 6, 83bitr4i 211 1 ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 1481  Vcvv 2689  cun 3073  wss 3075  {csn 3531  {cpr 3532
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538
This theorem is referenced by:  tpss  3692  prsspw  3699  exmidpw  6809
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