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Theorem prss 3646
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 3220 . 2 (({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶) ↔ ({𝐴} ∪ {𝐵}) ⊆ 𝐶)
2 prss.1 . . . 4 𝐴 ∈ V
32snss 3619 . . 3 (𝐴𝐶 ↔ {𝐴} ⊆ 𝐶)
4 prss.2 . . . 4 𝐵 ∈ V
54snss 3619 . . 3 (𝐵𝐶 ↔ {𝐵} ⊆ 𝐶)
63, 5anbi12i 455 . 2 ((𝐴𝐶𝐵𝐶) ↔ ({𝐴} ⊆ 𝐶 ∧ {𝐵} ⊆ 𝐶))
7 df-pr 3504 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
87sseq1i 3093 . 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 1465  Vcvv 2660  cun 3039  wss 3041  {csn 3497  {cpr 3498
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504
This theorem is referenced by:  tpss  3655  prsspw  3662  exmidpw  6770
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