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Theorem prss 3861
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 ((A C B C) ↔ {A, B} C)

Proof of Theorem prss
StepHypRef Expression
1 unss 3437 . 2 (({A} C {B} C) ↔ ({A} ∪ {B}) C)
2 prss.1 . . . 4 A V
32snss 3838 . . 3 (A C ↔ {A} C)
4 prss.2 . . . 4 B V
54snss 3838 . . 3 (B C ↔ {B} C)
63, 5anbi12i 678 . 2 ((A C B C) ↔ ({A} C {B} C))
7 df-pr 3742 . . 3 {A, B} = ({A} ∪ {B})
87sseq1i 3295 . 2 ({A, B} C ↔ ({A} ∪ {B}) C)
91, 6, 83bitr4i 268 1 ((A C B C) ↔ {A, B} C)
Colors of variables: wff setvar class
Syntax hints:  wb 176   wa 358   wcel 1710  Vcvv 2859  cun 3207   wss 3257  {csn 3737  {cpr 3738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259  df-sn 3741  df-pr 3742
This theorem is referenced by:  tpss  3871  prsspw  3878  uniintsn  3963
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